Unit 2: Exponential and Logarithmic Functions

Arithmetic and Geometric Sequences (Patterns Behind Linear vs. Exponential)

A sequence is an ordered list of numbers, where each number is a term. Sequences can be finite or infinite.

An arithmetic sequence is a sequence in which each successive term changes by a common difference, meaning a constant additive change (constant rate of change).

A geometric sequence is a sequence in which each successive term changes by a common ratio, meaning a constant multiplicative change (consistent proportional change).

This difference-vs.-ratio idea is one of the fastest ways to decide what kind of function model fits a situation. Over equal-length input intervals, if output values change at a constant rate (you keep adding the same amount), that points to a linear relationship. If output values change at a proportional rate (you keep multiplying by the same factor), that points to an exponential relationship. In many problems, you can determine the model type and parameters using just two distinct values from a sequence or two input-output points from a function.

Exam Focus

• Typical question patterns:
  • Identify whether data behave more like an arithmetic pattern (constant differences) or geometric pattern (constant ratios).
  • Use two values to determine a linear vs. exponential relationship.
• Common mistakes:
  • Checking differences when you should check ratios (or vice versa).
  • Forgetting that “proportional change” means multiplying by a constant factor, not adding a constant amount.

Exponential Functions: What They Are and How Their Graphs Behave

What an exponential function is (and why it’s different from a power function)

An exponential function is a function where the input variable appears in the exponent. The parent form is:

$f(x)=b^x$

Here, $b$ is a positive constant called the base.

This matters because exponential functions naturally model repeated multiplication: the change in a quantity is proportional to the current amount (population growth, interest, radioactive decay, etc.). By contrast, a power function like:

$g(x)=x^2$

has the variable in the base, not the exponent. That swap completely changes the behavior and graph shape.

Growth vs. decay (the role of the base)

For the parent function $b^x$, the base controls whether the function grows or decays.

$b>1$

means exponential growth.

$0<b<1$

means exponential decay.

You can see why by comparing consecutive inputs:

$b^{x+1}=b\cdot b^x$

Each step to the right multiplies the output by $b$.

Key anchor points (for any valid base):

$b^0=1$

so the graph passes through:

$\left(0,1\right)$

Also:

$b^1=b$

so the graph passes through:

$\left(1,b\right)$

Domain, range, asymptotes, monotonicity, and concavity

For $f(x)=b^x$ (with valid base), the features are:

• Domain: all real numbers.
• Range: outputs are always positive.
• Horizontal asymptote:

$y=0$

• End behavior for growth bases:

$\lim_{x\to\infty}b^x=\infty$

$\lim_{x\to-\infty}b^x=0$

• End behavior for decay bases:

$\lim_{x\to\infty}b^x=0$

$\lim_{x\to-\infty}b^x=\infty$

Exponential functions are always increasing or always decreasing (they’re one-to-one, so they have no local extrema unless you restrict to a closed interval). The parent exponential $b^x$ is always concave up. After transformations, a vertical reflection (multiplying by a negative) can make the graph concave down, and in either case exponential graphs have no inflection points.

Transformations: building more realistic models

A common transformed exponential model is:

$f(x)=a\cdot b^{x-h}+k$

• $a$ controls vertical stretch/compression and reflection.
  • If $|a|>1$, the graph stretches vertically.
  • If $0<|a|<1$, the graph compresses vertically.
  • If $a<0$, the graph reflects across the horizontal line:

$y=k$

• $h$ shifts horizontally (right by $h$ when the exponent is $x-h$).
• $k$ shifts vertically and moves the horizontal asymptote.

For transformed exponentials, the horizontal asymptote becomes:

$y=k$

and the range depends on the sign of $a$:

• If $a>0$, then outputs satisfy:

$f(x)>k$

• If $a<0$, then outputs satisfy:

$f(x)<k$

A frequent error is thinking $k$ changes the growth factor. It doesn’t. The multiplicative growth from left to right is still controlled by the base.

Two especially useful transformation identities

A horizontal translation in the exponent can be rewritten as a vertical scaling:

$b^{x+k}=b^x\cdot b^k$

So:

$f(x)=b^{x+k}=a\,b^x$

where:

$a=b^k$

A horizontal dilation/compression can be described by:

$f(x)=b^{cx}$

where $c\ne 0$. You can also interpret this as changing the effective base, since:

$b^{cx}=\left(b^c\right)^x$

and $b^c$ is a constant.

Reading the multiplicative rate of change

For:

$f(x)=a\cdot b^x$

the constant ratio over a one-unit increase in $x$ is:

$\frac{f(x+1)}{f(x)}=b$

If a quantity increases by $r$ percent per time period, then:

$b=1+\frac{r}{100}$

If it decreases by $r$ percent per time period, then:

$b=1-\frac{r}{100}$

Constructing exponential functions from data (and a key “shift” idea)

An exponential function can often be constructed from a common ratio and an initial value, or from two input-output pairs.

In real datasets, sometimes a constant must be added to or subtracted from the dependent variable to reveal a proportional growth pattern. In practice, that means you might recognize a model like:

$y=a\cdot b^x+k$

and check whether the values of:

$y-k$

have a roughly constant ratio.

Doubling every day vs. doubling every week

If $d$ is the number of days and a quantity doubles every day, a simple model is:

$f(d)=2^d$

If instead a quantity increases by a factor of 2 every 7 days, you want the exponent to count “number of weeks,” so:

$f(d)=2^{d/7}$

An equivalent way to write the “every 7 days” idea is:

$f(d)=\left(2^7\right)^{d/7}$

because the quantity increases by a factor of:

$2^7$

every 7 days.

Example 1: Interpret parameters from an exponential model

Suppose:

$P(t)=500\cdot 1.08^t$

where $t$ is in years.

• 500 is the initial amount because:

$P(0)=500\cdot 1.08^0=500$

• 1.08 is the growth factor per year, meaning 8% growth per year.

Example 2: Graph features from a transformed exponential

Given:

$f(x)=-2\cdot 3^{x-1}+5$

• Horizontal asymptote:

$y=5$

• Since $a=-2$ is negative, the graph is reflected across $y=5$, so it approaches 5 from below. A growth base reflected this way makes the function decrease as $x$ increases.
• A point on the graph: at $x=1$,

$f(1)=-2\cdot 3^0+5=3$

So:

$\left(1,3\right)$

is on the graph.

Exam Focus

• Typical question patterns:
  • Identify growth vs. decay and interpret parameters $a$, $b$, $h$, $k$ from context or a graph.
  • Determine asymptotes, domain/range, intercepts, and end behavior for a transformed exponential.
  • Compare two exponentials by analyzing their growth factors or initial values.
  • Use proportional change ideas (ratios) to justify an exponential model.
• Common mistakes:
  • Treating $b$ as an additive rate instead of a multiplicative factor.
  • Forgetting that vertical shifts move the horizontal asymptote to $y=k$.
  • Mixing up a horizontal shift with a vertical scaling (especially $b^{x+k}=b^x\cdot b^k$).
  • Claiming a parent exponential outputs negatives (it does not; negatives happen only after transformations).

Exponential Modeling in Context: Percent Change, Doubling/Half-Life, and Interest

Discrete-time vs. continuous-time models

In AP Precalculus, exponentials are models of repeated multiplication.

Discrete-time (changes happen in equal steps):

$A(t)=A_0\cdot b^t$

Continuous-time (changes happen continuously):

$A(t)=A_0\cdot e^{kt}$

Both are exponential models; they encode growth rate differently.

Building a model from an initial value and a percent rate

If an amount starts at $A_0$ and changes by $r$ percent per period:

$A(t)=A_0\left(1+\frac{r}{100}\right)^t$

For decay, you can treat $r$ as negative or use:

$1-\frac{r}{100}$

Example 1: Depreciation

A car worth 24000 dollars depreciates 15% per year.

Growth factor:

$b=1-0.15=0.85$

Model:

$V(t)=24000\cdot 0.85^t$

Interpretation: each year, the value is multiplied by 0.85.

Doubling time and half-life (solving for time)

If a quantity doubles every $d$ units of time:

$A(t)=A_0\cdot 2^{t/d}$

If a quantity has half-life $h$:

$A(t)=A_0\cdot \left(\frac{1}{2}\right)^{t/h}$

These forms make the “every $d$” or “every $h$” meaning immediate.

Example 2: Bacteria growth with doubling time

A culture starts with 300 bacteria and doubles every 4 hours:

$N(t)=300\cdot 2^{t/4}$

After 10 hours:

$N(10)=300\cdot 2^{10/4}=300\cdot 2^{2.5}$

Compound interest (discrete compounding)

If $P$ is principal, $r$ is annual interest rate (decimal), compounded $n$ times per year, then after $t$ years:

$A(t)=P\left(1+\frac{r}{n}\right)^{nt}$

The factor is the growth per compounding period, and the exponent counts the total number of compounding periods.

Example 3: Monthly compounding

With $P=5000$, $r=0.06$, monthly compounding means $n=12$:

$A(t)=5000\left(1+\frac{0.06}{12}\right)^{12t}$

Continuous growth (using the base $e$)

Continuous exponential growth/decay is modeled by:

$A(t)=A_0\cdot e^{kt}$

Growth vs. decay is determined by the sign of $k$.

Continuous interest is often written:

$A(t)=Pe^{rt}$

A common misconception is thinking $k$ is “a percent.” It’s a continuous rate per unit time. The relationship between a discrete factor $b$ and a continuous rate $k$ is:

$b=e^k$

and:

$k=\ln(b)$

Modeling from tables: constant ratios

A table suggests an exponential relationship if the ratio over equal input steps is constant (or close):

$\frac{f(x+\Delta x)}{f(x)}$

Example 4: Recognize exponential vs. linear from a table

If outputs go 50, 100, 200, 400 as $x$ increases by 1, the ratio is 2 each step, so:

$f(x)=50\cdot 2^x$

Exam Focus

• Typical question patterns:
  • Build an exponential model from “starts at” and “grows/decays by” language.
  • Use doubling time or half-life descriptions to write a function.
  • Apply compound interest formulas and interpret what each parameter means.
  • Identify exponential behavior from tables using ratios.
• Common mistakes:
  • Mixing up the doubling-time form (using $2^{dt}$ instead of $2^{t/d}$).
  • Using percent as a whole number (using 8 instead of 0.08).
  • Forgetting the exponent counts total compounding periods $nt$.

Solving Exponential Equations (Without and With Logarithms)

The big idea

Solving an exponential equation means finding inputs that make two exponential expressions equal. If you can rewrite both sides using the same base, the equation becomes much simpler because exponentials with the same base are one-to-one.

Method 1: Rewrite with a common base

If:

$a^x=a^y$

(with a valid base), then:

$x=y$

Example 1: Common base

Solve:

$3^{2x-1}=27$

Rewrite:

$27=3^3$

So:

$3^{2x-1}=3^3$

Set exponents equal:

$2x-1=3$

$2x=4$

$x=2$

Method 2: Use substitution when bases are related

When you see expressions like $4^x$ and $2^x$ together, rewrite with a common base and substitute.

Example 2: Substitution

Solve:

$4^x-5\cdot 2^x+6=0$

Rewrite:

$4^x=\left(2^2\right)^x=2^{2x}$

So:

$2^{2x}-5\cdot 2^x+6=0$

Let:

$u=2^x$

Then:

$2^{2x}=\left(2^x\right)^2=u^2$

Substitute:

$u^2-5u+6=0$

Factor:

$\left(u-2\right)\left(u-3\right)=0$

So:

$u=2$

or:

$u=3$

Undo the substitution:

$2^x=2$

gives:

$x=1$

And:

$2^x=3$

has a real solution, but it’s not an integer; you express it with logarithms or approximate with technology.

Method 3: Solve using logarithms

When you can’t match bases, take logs of both sides and use the power rule.

Example:

$2^x=7$

Take logs:

$\ln\left(2^x\right)=\ln\left(7\right)$

Bring down the exponent:

$x\ln\left(2\right)=\ln\left(7\right)$

Solve:

$x=\frac{\ln\left(7\right)}{\ln\left(2\right)}$

This is also:

$x=\log_2\left(7\right)$

Solving with graphs (conceptual method)

Graphing helps you check whether solutions exist and estimate them. For example, to solve $2^x=7$, you can graph the intersection of:

$y=2^x$

and:

$y=7$

or graph:

$y=2^x-7$

and find the zero.

Exam Focus

• Typical question patterns:
  • Solve exponential equations by rewriting with a common base.
  • Use substitution to turn an exponential equation into a quadratic-like equation.
  • Estimate solutions using intersections or zeros on graphs.
• Common mistakes:
  • Setting exponents equal when bases are not the same.
  • Forgetting non-integer real solutions exist (like $2^x=3$).
  • Algebra mistakes when rewriting forms like $b^{2x}=\left(b^x\right)^2$.

Composition, Identity, and Inverse Functions (Including Exponential–Log Inverses)

Composition of functions and domain restrictions

The composition:

$(f\circ g)(x)=f\left(g(x)\right)$

means the outputs of $g$ become inputs of $f$. The **domain of a composite** is restricted to values of $x$ where:

1. $g(x)$ is defined, and
2. $g(x)$ lands inside the domain of $f$.

Typically:

$f\left(g(x)\right)\ne g\left(f(x)\right)$

so $f\circ g$ and $g\circ f$ are different functions.

Additive vs. multiplicative transformations

Thinking in terms of transformations helps across this whole unit.

• Additive transformations correspond to translations, such as:

$g(x)=x+k$

• Multiplicative transformations correspond to dilations (scaling), such as:

$g(x)=kx$

Identity function

The identity function is:

$f(x)=x$

Composing with the identity leaves a function unchanged:

$g(f(x))=g(x)$

and:

$f(g(x))=g(x)$

The identity function plays a role similar to 0 (additive identity) when adding and 1 (multiplicative identity) when multiplying.

Inverse functions

An inverse function reverses a mapping: if:

$f(a)=b$

then:

$f^{-1}(b)=a$

When inverses exist, composing a function and its inverse gives the identity:

$f\left(f^{-1}(x)\right)=x$

and:

$f^{-1}\left(f(x)\right)=x$

Also, a function’s domain and range swap with its inverse’s range and domain (and in some cases, the domain may need to be restricted to make an inverse a function).

Inverse of exponential functions: logarithms

Exponential functions and logarithmic functions are inverses and their graphs reflect over:

$y=x$

If:

$g(x)=b^x$

then:

$g^{-1}(x)=\log_b(x)$

and the inverse relationships show up through composition:

$\log_b\left(b^x\right)=x$

and, for positive inputs:

$b^{\log_b(x)}=x$

This also connects to how the “growth patterns” flip when you invert. Exponential growth means outputs change multiplicatively as inputs change additively; logarithmic growth means outputs change additively as inputs change multiplicatively.

Exam Focus

• Typical question patterns:
  • Compute and interpret compositions and identify the correct domain of a composite.
  • Use inverse-function ideas to justify why logs solve for exponents.
  • Identify when a domain restriction is needed for an inverse to be a function.
• Common mistakes:
  • Forgetting that the domain of $f\circ g$ must keep $g(x)$ inside the domain of $f$.
  • Assuming $f\circ g=g\circ f$.
  • Mixing up which quantities swap between a function and its inverse (domain/range).

Logarithms: The Inverse of an Exponential

Why logarithms exist

Exponentials answer “given the input, what is the output?” In modeling you often need the inverse question: “given the output, what input produced it?” Solving for a variable in an exponent leads directly to logarithms.

Definition (exponential form and log form)

For a valid base and positive input:

$\log_b(x)=y$

means exactly:

$b^y=x$

The logarithm is the exponent you put on base $b$ to get $x$.

Domain, range, asymptotes, and general behavior

For:

$f(x)=\log_b(x)$

• Domain:

$x>0$

• Range: all real numbers.
• Vertical asymptote:

$x=0$

Log functions are one-to-one, so they are always increasing or always decreasing (depending on the base and transformations), and they do not have extrema except on closed intervals.

For the parent log function, concavity depends on the base:

$b>1$

gives an increasing log that is concave down.

$0<b<1$

gives a decreasing log that is concave up.

Multiplying by a negative reflects the graph and flips concavity. In all cases, logarithmic graphs have no inflection points.

End behavior is unbounded: logs grow without bound in one direction and decrease without bound in the other.

Common bases and notation

• General logarithm:

$\log_b(x)$

• Common logarithm (typically base 10 in AP contexts):

$\log(x)$

• Natural logarithm (base $e$):

$\ln(x)$

Key inverse pairs and symmetry

Since exponentials and logs are inverses, their graphs reflect across:

$y=x$

Also:

$\log_b(1)=0$

because:

$b^0=1$

and:

$\log_b(b)=1$

because:

$b^1=b$

Constructing a logarithmic function (data/conditions)

A logarithmic function can be constructed from a proportion and a real zero, or from two input-output pairs, using the idea that logs reverse exponentiation.

A useful way to think about a log graph is that each increase of 1 in the output corresponds to multiplying the input by the base. For example, in a base-10 log, increasing the output by 1 corresponds to multiplying the input by 10.

Another way to recognize logarithmic behavior is the “inverse of exponential” idea: for a logarithmic function, equal increments in output correspond to proportional (multiplicative) changes in input. A caution worth remembering is that a proportional-input pattern over equal output intervals strongly suggests logarithmic behavior, but this recognition rule is not a reversible “if and only if” test in messy real-world data.

Change of base

To evaluate logs with bases your calculator doesn’t directly provide:

$\log_b(x)=\frac{\ln(x)}{\ln(b)}$

and equivalently:

$\log_b(x)=\frac{\log(x)}{\log(b)}$

Example 1: Evaluate using change of base

Compute:

$\log_2(7)$

Using natural log:

$\log_2(7)=\frac{\ln(7)}{\ln(2)}$

Example 2: Convert between exponential and log forms

Rewrite:

$5^x=120$

in logarithmic form:

$x=\log_5(120)$

Exam Focus

• Typical question patterns:
  • Convert between $b^y=x$ and $\log_b(x)=y$.
  • Identify domain/range/asymptotes for log functions and their transformations.
  • Use change of base to evaluate expressions or solve equations.
• Common mistakes:
  • Allowing negative or zero inputs into a logarithm.
  • Confusing $\log_b(x)$ with multiplication.
  • Forgetting log outputs can be negative.

Properties of Logarithms and How to Rewrite Expressions

Why log properties matter

Log properties let you turn multiplication into addition and division into subtraction, which is especially useful for solving equations with variables in exponents and for linearizing exponential patterns.

The three core log properties

Assume valid base and positive inputs.

Product rule:

$\log_b(MN)=\log_b(M)+\log_b(N)$

Quotient rule:

$\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$

Power rule:

$\log_b\left(M^p\right)=p\log_b(M)$

There is no property for splitting a sum:

$\log_b(M+N)$

cannot be rewritten as a sum of logs.

Expanding vs. condensing

Expanding rewrites one log as multiple logs; condensing rewrites multiple logs as a single log.

Example 1: Expand

Expand:

$\log_3\left(\frac{9x^2}{\sqrt{y}}\right)$

Rewrite division as subtraction:

$\log_3(9x^2)-\log_3\left(\sqrt{y}\right)$

Rewrite product as addition:

$\log_3(9)+\log_3\left(x^2\right)-\log_3\left(\sqrt{y}\right)$

Use the power rule and $\sqrt{y}=y^{1/2}$:

$\log_3(9)+2\log_3(x)-\frac{1}{2}\log_3(y)$

(And $\log_3(9)=2$ because $3^2=9$.)

Example 2: Condense

Condense:

$2\ln(x)-\ln(3)+\frac{1}{2}\ln(y)$

Move coefficients up as exponents:

$\ln\left(x^2\right)-\ln(3)+\ln\left(y^{1/2}\right)$

Combine addition as multiplication:

$\ln\left(x^2\cdot y^{1/2}\right)-\ln(3)$

Combine subtraction as division:

$\ln\left(\frac{x^2\cdot y^{1/2}}{3}\right)$

Transformations of logarithmic graphs

A transformed logarithmic function often looks like:

$f(x)=a\cdot \log_b(x-h)+k$

The vertical asymptote is where the argument becomes zero:

$x=h$

because the input must satisfy:

$x-h>0$

Example 3: Identify the asymptote and domain

For:

$f(x)=\ln(x-4)+2$

Domain comes from:

$x-4>0$

so:

$x>4$

and the vertical asymptote is:

$x=4$

Exam Focus

• Typical question patterns:
  • Expand or condense expressions using product/quotient/power rules.
  • Identify domain restrictions created by log arguments.
  • Analyze transformations of log graphs (shifts, stretches, asymptotes).
• Common mistakes:
  • Applying log rules to sums.
  • Ignoring positivity requirements (logs require positive inputs).
  • Forgetting a horizontal shift changes the vertical asymptote to $x=h$.

Solving Equations Involving Logarithms (and Checking Solutions)

The basic strategy

Usually you isolate a logarithm (or condense multiple logs into one), convert to exponential form, and solve. Two habits matter every time:

1. Every log argument must be positive.
2. Check solutions in the original equation to avoid extraneous solutions.

Type A: Single logarithm equals a number

If:

$\log_b(X)=c$

then:

$X=b^c$

Example 1

Solve:

$\log_5(2x-1)=3$

Convert to exponential form:

$2x-1=5^3$

$2x-1=125$

$2x=126$

$x=63$

Domain check requires:

$2x-1>0$

so $x=63$ is valid.

Type B: Log equals log (one-to-one property)

If bases match and logs are defined, then:

$\log_b(M)=\log_b(N)$

implies:

$M=N$

Example 2

Solve:

$\ln(x-2)=\ln(10-x)$

Set arguments equal:

$x-2=10-x$

$2x=12$

$x=6$

Domain check:

$x-2>0$

and:

$10-x>0$

so $x=6$ is valid.

Type C: Multiple logs (condense, then convert)

Example 3

Solve:

$\log_2(x)+\log_2(x-2)=3$

Domain restrictions:

$x>0$

and:

$x-2>0$

so:

$x>2$

Condense:

$\log_2\left(x(x-2)\right)=3$

Convert:

$x(x-2)=2^3$

$x^2-2x=8$

$x^2-2x-8=0$

Factor:

$\left(x-4\right)\left(x+2\right)=0$

So:

$x=4$

or:

$x=-2$

The restriction $x>2$ eliminates $x=-2$, leaving $x=4$.

Logarithmic inequalities (conceptual caution)

Inequalities require attention to monotonicity. If the base is greater than 1, the log function is increasing and inequality direction is preserved. If the base is between 0 and 1, the log function is decreasing and inequality direction reverses.

Exam Focus

• Typical question patterns:
  • Solve log equations by converting between log and exponential forms.
  • Solve equations with multiple logs by condensing first.
  • Identify extraneous solutions created by domain restrictions.
• Common mistakes:
  • Forgetting domain restrictions and accepting invalid solutions.
  • Condensing with the wrong sign in quotient situations.
  • Assuming $\ln(A)=\ln(B)$ implies $A=B$ without ensuring both sides can be positive.

Using Logarithms to Solve Exponential Equations and Interpret Models

Solving exponential equations with logs (standard technique)

When the variable is in the exponent and bases can’t be matched, logs are the tool.

General approach:

1. Isolate the exponential expression.
2. Take a log of both sides.
3. Use the power rule to bring the exponent down.
4. Solve the resulting linear equation.

Example 1

Solve:

$3^{2x+1}=50$

Take natural logs:

$\ln\left(3^{2x+1}\right)=\ln(50)$

Bring down the exponent:

$\left(2x+1\right)\ln(3)=\ln(50)$

Solve:

$2x+1=\frac{\ln(50)}{\ln(3)}$

$2x=\frac{\ln(50)}{\ln(3)}-1$

$x=\frac{1}{2}\left(\frac{\ln(50)}{\ln(3)}-1\right)$

Interpreting continuous growth rates and solving for time

For:

$A(t)=A_0e^{kt}$

solving for the time to reach a target $A_T$ gives:

$A_T=A_0e^{kt}$

$\frac{A_T}{A_0}=e^{kt}$

$\ln\left(\frac{A_T}{A_0}\right)=kt$

$t=\frac{1}{k}\ln\left(\frac{A_T}{A_0}\right)$

Example 2: Time to reach a threshold

A medication amount decays continuously with:

$A(t)=120e^{-0.4t}$

When does it drop to 30?

$30=120e^{-0.4t}$

$\frac{1}{4}=e^{-0.4t}$

$\ln\left(\frac{1}{4}\right)=-0.4t$

$t=\frac{\ln\left(\frac{1}{4}\right)}{-0.4}$

Connecting discrete and continuous models

A discrete model:

$A(t)=A_0b^t$

can be rewritten in base $e$ using:

$b^t=e^{t\ln(b)}$

So:

$A(t)=A_0e^{\ln(b)\cdot t}$

which shows the equivalent continuous rate:

$k=\ln(b)$

Exam Focus

• Typical question patterns:
  • Solve equations like $a\cdot b^{cx+d}=m$ using logarithms.
  • Solve for time in growth/decay contexts, especially with $e^{kt}$ models.
  • Convert between $A_0b^t$ and $A_0e^{kt}$.
• Common mistakes:
  • Not isolating the exponential before taking logs.
  • Misusing the power rule (for example treating $\ln\left(b^x\right)$ incorrectly).
  • Sign mistakes when solving for time in decay (negative $k$).

Modeling with Exponentials and Logs from Data: Regression, Linearization, Semi-Log Plots, and Model Validation

Recognizing exponential patterns in data

A dataset suggests exponential behavior if equal steps in the input correspond to roughly constant ratios in the output. Linear patterns show constant differences; exponential patterns show constant ratios. Real data may be noisy, so you look for a trend rather than perfect consistency.

Exponential regression

Exponential regression fits a model of the form:

$y=a\cdot b^x$

• $a$ approximates the value at $x=0$.
• $b$ is the multiplicative factor per unit increase in $x$.

Regression gives a best fit, not an exact relationship, so interpretations should use “approximately” and should respect a reasonable input interval.

Example: Interpret parameters from an exponential regression

If regression yields:

$y=120\cdot 1.15^x$

then 120 is the predicted value at $x=0$, and each increase of 1 in $x$ multiplies the predicted value by 1.15 (about 15% growth per unit).

Linearizing an exponential model using logarithms

Starting with:

$y=a\cdot b^x$

Take natural logs:

$\ln(y)=\ln\left(a\cdot b^x\right)$

Use properties:

$\ln(y)=\ln(a)+\ln\left(b^x\right)$

$\ln(y)=\ln(a)+x\ln(b)$

So plotting $\ln(y)$ versus $x$ should look linear if an exponential model is appropriate. The slope is $\ln(b)$ and the intercept is $\ln(a)$, meaning:

$b=e^{\text{slope}}$

$a=e^{\text{intercept}}$

A common mistake is thinking the slope is $b$; on a log-transformed scale it is $\ln(b)$.

Semi-log plots

A semi-log plot has a logarithmically scaled y-axis and a linearly scaled x-axis. It is used to visualize exponential functions, because exponentials appear as straight-ish lines on a semi-log plot.

On a semi-log plot, you typically do not need to add a constant to the dependent variable to reveal an exponential pattern (whereas in raw data, a vertical shift might hide constant ratios until you adjust by subtracting a constant).

A general linear relationship on a semi-log plot can be written as:

$\log_n(y)=mx+c$

with a valid log base:

$n>0$

and:

$n\ne 1$

This corresponds to an exponential model:

$y=n^{mx+c}$

which can be rewritten as:

$y=\left(n^c\right)\left(n^m\right)^x$

So the “initial linear value” is the intercept $c$ on the $\log_n(y)$ scale, and the “linear rate of change” is the slope $m$ on that same scale.

Given two data points, the slope on the semi-log plot can be computed by:

$m=\frac{\log_n(y_2)-\log_n(y_1)}{x_2-x_1}$

and then the intercept by:

$c=\log_n(y_1)-mx_1$

Competing function model validation

When deciding between competing models (for example, linear vs. exponential vs. logarithmic), a model may be considered appropriate if the residual pattern is not systematic (the regression errors show no clear pattern).

A practical check is comparing predicted vs. actual values and describing the error. Depending on the context and interval, it may be acceptable (or even preferred) for a model to consistently overestimate or underestimate, but you should say so explicitly and justify it.

Logarithmic models (when they make sense)

A logarithmic model often looks like:

$y=a+b\ln(x)$

This can fit situations where growth is fast initially and then slows (diminishing returns, learning curves, etc.). Remember domain restrictions: log models require positive inputs (or greater than a horizontal shift).

Interpreting model reasonableness and domain

In context you should restrict to meaningful inputs (often $t\ge 0$ for time, and outputs like populations should not be negative). Also, log-based methods require positive values inside the log; for example, you cannot compute $\ln(y)$ if $y\le 0$.

Exam Focus

• Typical question patterns:
  • Decide whether a dataset is better modeled as linear or exponential (differences vs. ratios).
  • Interpret regression parameters $a$ and $b$ in context.
  • Use log transformations or semi-log plots to justify an exponential model.
  • Evaluate competing models using predicted vs. actual values and residual/error reasoning.
• Common mistakes:
  • Extrapolating far outside the data range without justification.
  • Interpreting $b$ as an additive increase rather than a multiplicative factor.
  • Forgetting log transformations require positive values (you cannot take logs of non-positive numbers).
  • Assuming “looks linear” on a semi-log plot guarantees an exponential model without considering context and domain.

Combining Exponentials and Logarithms: Inverses, Compositions, and Function Behavior

Inverses and composition identities

Because logs and exponentials undo each other, the key identities are:

$\log_b\left(b^x\right)=x$

and for positive inputs:

$b^{\log_b(x)}=x$

The domain condition matters in the second identity because logs only accept positive inputs.

Simplifying expressions with $e$ and $\ln$

Special cases with base $e$ show up constantly:

$\ln\left(e^x\right)=x$

and for positive inputs:

$e^{\ln(x)}=x$

Example 1: Simplify

Simplify:

$\ln\left(e^{3x-2}\right)$

Result:

$3x-2$

Example 2: Simplify with domain awareness

Simplify:

$e^{\ln(5x-1)}$

Result:

$5x-1$

with the required condition:

$5x-1>0$

Transformations and asymptotes (a common mix-up)

Exponentials have horizontal asymptotes that shift with vertical translations.

Logarithms have vertical asymptotes that shift with horizontal translations.

For example:

$f(x)=\log_2(x-1)+3$

has a vertical asymptote at:

$x=1$

while:

$g(x)=2\cdot 10^{-x}+1$

has a horizontal asymptote at:

$y=1$

Function operations and restrictions (especially with logs)

Whenever logs appear, enforce positivity of each argument.

Example:

$h(x)=\ln(x-2)+\ln(5-x)$

requires:

$x-2>0$

and:

$5-x>0$

so the domain is:

$2<x<5$

Exam Focus

• Typical question patterns:
  • Simplify expressions using inverse relationships.
  • Determine domains of functions built from log expressions.
  • Analyze asymptotes and shifts for transformed exponential and log functions.
• Common mistakes:
  • Canceling logs and exponentials with mismatched bases.
  • Ignoring domain restrictions after simplifying.
  • Confusing which type of asymptote belongs to exponentials vs. logarithms.