Exponential and Logarithmic Functions

Exponential Functions

• Definition of Exponential Function: The exponential function with base $a$ is defined for all real numbers $x$ by:

$f(x) = a^x$

  where $a > 0$ and $a \neq 1$.

• Characteristics and Rate of Growth: Exponential functions feature an independent variable in the exponent. To illustrate how rapidly these function values increase, consider $f(x) = 2^x$:

  • $f(3) = 2^3 = 8$
  • $f(10) = 2^{10} = 1024$
  • $f(30) = 2^{30} = 1,073,741,824$

• Irrational Exponents: To define $a^x$ when $x$ is an irrational number, $x$ is approximated by rational numbers.

  • For example, since $\sqrt{3} \approx 1.73205\dots$ is an irrational number, $a^{\sqrt{3}}$ is successively approximated using rational powers:

$a^{1.7}, a^{1.73}, a^{1.732}, a^{1.7320}, a^{1.73205}, \dots$

• Evaluating $5^{\sqrt{3}}$ using calculator approximations:

$5^{\sqrt{3}} \approx 5^{1.732} \approx 16.2411\dots$

• Using more decimal places of $\sqrt{3}$ yields a more precise approximation of $5^{\sqrt{3}}$.

  • Evaluating Exponential Functions:

• For $f(x) = 3^x$, specific function evaluations yield:

  • $f(5) = 3^5 = 243$ (Calculator keystrokes: 3 ^ 5 ENTER ; Output: 243)

  • $f\left(-\frac{2}{3}\right) = 3^{-2/3} \approx 0.4807$ (Calculator keystrokes: 3 ^ ( (-) 2 ÷ 3 ) ENTER ; Output: 0.4807498)

  • $f(\pi) = 3^{\pi} \approx 31.544$ (Calculator keystrokes: 3 ^ π ENTER ; Output: 31.5442807)

  • $f(\sqrt{2}) = 3^{\sqrt{2}} \approx 4.7288$ (Calculator keystrokes: 3 ^ √ 2 ENTER ; Output: 4.7288043)

  • Graphs of Exponential Functions:

• Graphing $f(x) = 3^x$ and $g(x) = \left(\frac{1}{3}\right)^x$ by plotting points:

  | $x$ | $f(x) = 3^x$ | $g(x) = \left(\frac{1}{3}\right)^x$ |   | :---: | :---: | :---: |   | $-3$ | $\frac{1}{27}$ | $27$ |   | $-2$ | $\frac{1}{9}$ | $9$ |   | $-1$ | $\frac{1}{3}$ | $3$ |   | $0$ | $1$ | $1$ |   | $1$ | $3$ | $\frac{1}{3}$ |   | $2$ | $9$ | $\frac{1}{9}$ |   | $3$ | $27$ | $\frac{1}{27}$ |

Compound Interest

• Derivation of the Compound Interest Formula:
  • If an amount of money $P$ (the principal) is invested at an interest rate $i$ per time period, the interest after one time period is $Pi$, and the total amount $A$ is:

$A = P + Pi = P(1 + i)$

• If the interest is reinvested, the new principal is $P(1 + i)$. The amount after a second time period is:

$A = P(1 + i)(1 + i) = P(1 + i)^2$

• After a third time period, the amount is:

$A = P(1 + i)^3$

• In general, after $k$ time periods, the amount is:

$A = P(1 + i)^k$

    This represents an exponential function with base $(1 + i)$.

• General Compound Interest Formula:
  • If the annual interest rate is $r$ and interest is compounded $n$ times per year, the rate per time period is $i = \frac{r}{n}$, and the number of periods in $t$ years is $nt$:

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

• Parameters:

  • $A(t)$ = amount after $t$ years

  • $P$ = principal

  • $r$ = annual interest rate per year

  • $n$ = number of times interest is compounded per year

  • $t$ = number of years

  • Example Comparison of Compounding Frequencies:

• Sum invested: $P = \$1000$, annual interest rate: $r = 0.12$ ($12\%$), time period: $t = 3$ years.

  | Compounding | $n$ | Amount after 3 years |   | :--- | :---: | :--- |   | Annual | $1$ | $1000\left(1 + \frac{0.12}{1}\right)^{1(3)} = \$1404.93$ |   | Semiannual | $2$ | $1000\left(1 + \frac{0.12}{2}\right)^{2(3)} = \$1418.52$ |   | Quarterly | $4$ | $1000\left(1 + \frac{0.12}{4}\right)^{4(3)} = \$1425.76$ |   | Monthly | $12$ | $1000\left(1 + \frac{0.12}{12}\right)^{12(3)} = \$1430.77$ |   | Daily | $365$ | $1000\left(1 + \frac{0.12}{365}\right)^{365(3)} = \$1433.24$ |

The Natural Exponential Function

• Definition of the Number $e$:
  • The number $e$ is defined as the limiting value that the expression $\left(1 + \frac{1}{n}\right)^n$ approaches as $n$ becomes arbitrarily large.

  | $n$ | $\left(1 + \frac{1}{n}\right)^n$ |   | :---: | :---: |   | $1$ | $2.00000$ |   | $5$ | $2.48832$ |   | $10$ | $2.59374$ |   | $100$ | $2.70481$ |   | $1000$ | $2.71692$ |   | $10,000$ | $2.71815$ |   | $100,000$ | $2.71827$ |   | $1,000,000$ | $2.71828$ |

• Rounded to 5 decimal places: $e \approx 2.71828$.

• Rounded to 20 decimal places: $e \approx 2.71828182845904523536$.

• The number $e$ is an irrational number; its exact value cannot be written in decimal form.

  • The Natural Exponential Function:

• Defined as the exponential function with base $e$:

$f(x) = e^x$

• Since $2 < e < 3$, the graph of $y = e^x$ lies strictly between the graphs of $y = 2^x$ and $y = 3^x$

• Exponential Model for the Spread of a Virus:
  • In a city of population $10,000$, the number of infected people after $t$ days is modeled by:

$v(t) = \frac{10,000}{5 + 1245e^{-0.97t}}$

• Initial infected count ($t = 0$):

$v(0) = \frac{10,000}{5 + 1245e^0} = \frac{10,000}{1250} = 8\text{ people}$

• Infections over time:
  • After 1 day ($t = 1$): $21\text{ people}$
  • After 2 days ($t = 2$): $54\text{ people}$
  • After 5 days ($t = 5$): $678\text{ people}$
• Graphical behavior: The number of infected individuals rises slowly initially, increases rapidly between Day 3 and Day 8, and then levels off near $2000\text{ people}$.

• Continuously Compounded Interest:
  • As the compounding frequency $n \to \infty$, let $m = \frac{n}{r}$. The compound interest formula transforms as follows:

$A(t) = P\left(1 + \frac{r}{n}\right)^{nt} = P\left[\left(1 + \frac{r}{n}\right)^{n/r}\right]^{rt} = P\left[\left(1 + \frac{1}{m}\right)^m\right]^{rt}$

• Since $\left(1 + \frac{1}{m}\right)^m \to e$ as $m \to \infty$, the amount compounded at "every instant" is given by:

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

• Example: For $P = \$1000$, $r = 0.12$ ($12\%$), and $t = 3$ years compounded continuously:

$A(3) = 1000e^{(0.12)(3)} = 1000e^{0.36} \approx \$1433.33$

Logarithmic Functions

• Definition of Logarithmic Functions:
  • Every exponential function $f(x) = a^x$ ($a > 0, a \neq 1$) passes the Horizontal Line Test and is one-to-one, possessing an inverse function $f^{-1}$.
  • The inverse function $f^{-1}$ is the logarithmic function with base $a$, denoted $\log_a$:

$f^{-1}(x) = y \iff f(y) = x$

$\log_a x = y \iff a^y = x$

• $\log_a x$ is the exponent to which the base $a$ must be raised to produce $x$.

  • Equivalence of Forms:

• Logarithmic form: $\log_a x = y$

• Exponential form: $a^y = x$

• Both forms share the exact same base $a$.

  | Logarithmic Form | Exponential Form |   | :--- | :--- |   | $\log_{10} 100,000 = 5$ | $10^5 = 100,000$ |   | $\log_2 8 = 3$ | $2^3 = 8$ |   | $\log_2\left(\frac{1}{8}\right) = -3$ | $2^{-3} = \frac{1}{8}$ |   | $\log_5 s = r$ | $5^r = s$ |

• Evaluating Logarithms Examples:

  • $\log_{10} 1000 = 3$ because $10^3 = 1000$
  • $\log_2 32 = 5$ because $2^5 = 32$
  • $\log_{10} 0.1 = -1$ because $10^{-1} = 0.1$
  • $\log_{16} 4 = \frac{1}{2}$ because $16^{1/2} = 4$

• Properties of Logarithms:

  1. $\log_a 1 = 0$ (Reason: $a^0 = 1$)
  2. $\log_a a = 1$ (Reason: $a^1 = a$)
  3. $\log_a(a^x) = x$ for all $x \in \mathbb{R}$ (Reason: $a$ must be raised to power $x$ to get $a^x$)
  4. $a^{\log_a x} = x$ for all $x > 0$ (Reason: $\log_a x$ is the power to which $a$ must be raised to get $x$)

• Graphs, Domain, and Range of Logarithmic Functions:

  • For $f(x) = a^x$ ($a > 1$), Domain = $\mathbb{R}$ and Range = $(0, \infty)$.
  • For $f^{-1}(x) = \log_a x$, Domain = $(0, \infty)$ and Range = $\mathbb{R}$.
  • The graph of $y = \log_a x$ is obtained by reflecting $y = a^x$ across the line $y = x$.
  • The $x$-intercept is $1$ (since $\log_a 1 = 0$).
  • The $y$-axis ($x = 0$) is a vertical asymptote because $\log_a x \to -\infty$ as $x \to 0^+$.

• Common Logarithms:
  • The logarithm with base $10$ is called the common logarithm and is denoted by omitting the base:

$\log x = \log_{10} x$

• Key values: $\log 10 = 1$, $\log 100 = 2$.

• Approximating $\log 50$: Since $10^1 = 10$ and $10^2 = 100$, $1 < \log 50 < 2$.

  • Logarithmic Scales Application (Sound Loudness):

• Loudness $B$ in decibels (dB) of a sound with physical intensity $I$ (in $\text{W/m}^2$) is modeled by:

$B = 10\log\left(\frac{I}{I_0}\right)$

    where $I_0$ is the intensity of a barely audible sound.

• Decibel level for a sound with intensity $I = 100I_0$:

$B = 10\log\left(\frac{100I_0}{I_0}\right) = 10\log(100) = 10 \times 2 = 20\text{ dB}$

• Natural Logarithms:
  • Logarithm with base $e$ is called the natural logarithm and is denoted by $\ln$:

$\ln x = \log_e x$

• Inverse relationship:

$\ln x = y \iff e^y = x$

• Properties of Natural Logarithms:
  1. $\ln 1 = 0$
  2. $\ln e = 1$
  3. $\ln e^x = x$
  4. $e^{\ln x} = x$

• Finding the Domain of a Logarithmic Function Example:
  • Find the domain of $f(x) = \ln(4 - x^2)$.
  • Solution: $\ln x$ is defined only when its input is strictly positive:

$\{x \mid 4 - x^2 > 0\} = \{x \mid x^2 < 4\} = \{x \mid |x| < 2\} = \{x \mid -2 < x < 2\} = (-2, 2)$

Laws of Logarithms

• Laws of Logarithms:

  • Let $a > 0, a \neq 1$, and let $A > 0, B > 0$.
  1. Product Law: $\log_a(AB) = \log_a A + \log_a B$
  • Description: The logarithm of a product is the sum of the logarithms of the factors.
  1. Quotient Law: $\log_a\left(\frac{A}{B}\right) = \log_a A - \log_a B$
  • Description: The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.
  1. Power Law: $\log_a(A^C) = C\log_a A$
  • Description: The logarithm of a power of a number is the exponent times the logarithm of the number.

• Proofs of the Laws of Logarithms:

  • Proof of Law 1: Let $u = \log_a A$ and $v = \log_a B$. In exponential form, $a^u = A$ and $a^v = B$. Thus:

$\log_a(AB) = \log_a(a^u a^v) = \log_a(a^{u+v}) = u + v = \log_a A + \log_a B$

• Proof of Law 2: Using Law 1:

$\log_a A = \log_a\left[\left(\frac{A}{B}\right)B\right] = \log_a\left(\frac{A}{B}\right) + \log_a B$

$\log_a\left(\frac{A}{B}\right) = \log_a A - \log_a B$

• Proof of Law 3: Let $u = \log_a A$, so $a^u = A$. Thus:

$\log_a(A^C) = \log_a\left((a^u)^C\right) = \log_a(a^{uC}) = uC = C\log_a A$

• Evaluating Expressions Using Laws of Logarithms Examples:

  • $\log_4 2 + \log_4 32 = \log_4(2 \cdot 32) = \log_4 64 = 3$ (since $4^3 = 64$)
  • $\log_2 80 - \log_2 5 = \log_2\left(\frac{80}{5}\right) = \log_2 16 = 4$ (since $2^4 = 16$)
  • $-\frac{1}{3}\log 8 = \log(8^{-1/3}) = \log\left(\frac{1}{2}\right) \approx -0.301$

• Expanding Logarithmic Expressions Examples:

  • $\log_2(6x) = \log_2 6 + \log_2 x$
  • $\log_5(x^3 y^6) = \log_5 x^3 + \log_5 y^6 = 3\log_5 x + 6\log_5 y$
  • $\ln\left(\frac{ab}{\sqrt[3]{c}}\right) = \ln(ab) - \ln \sqrt[3]{c} = \ln a + \ln b - \ln(c^{1/3}) = \ln a + \ln b - \frac{1}{3}\ln c$

• Psychological Application (The Law of Forgetting):

  • Formulated by Hermann Ebbinghaus (1850–1909):

$\log P = \log P_0 - c\log(t + 1)$

    where $P_0$ is the initial performance score, $P$ is the retained performance level after $t$ months, and $c$ is a task-dependent constant.

• Solving for $P$ algebraically:

$\log P = \log P_0 - \log(t + 1)^c$

$\log P = \log\left(\frac{P_0}{(t + 1)^c}\right)$

$P = \frac{P_0}{(t + 1)^c}$

• Calculations for $P_0 = 90$ and $c = 0.2$:
  • After 2 months ($t = 2$):

$P = \frac{90}{(2 + 1)^{0.2}} = \frac{90}{3^{0.2}} \approx 72$

- After 1 year ($t = 12$):

$P = \frac{90}{(12 + 1)^{0.2}} = \frac{90}{13^{0.2}} \approx 54$

• Change of Base Formula:
  • Derivation: To convert $\log_b x$ to base $a$, let $y = \log_b x$. Express in exponential form $b^y = x$, take $\log_a$ of both sides, apply Law 3, and solve for $y$:

$\log_a(b^y) = \log_a x$

$y\log_a b = \log_a x$

$y = \frac{\log_a x}{\log_a b}$

• Formula:

$\log_b x = \frac{\log_a x}{\log_a b}$

• Special Case ($x = a$):

$\log_b a = \frac{1}{\log_a b}$

• Change of Base Examples:
  • Evaluate $\log_8 5$ using common logarithms ($a = 10$):

$\log_8 5 = \frac{\log_{10} 5}{\log_{10} 8} \approx 0.77398$

- Evaluate $\log_9 20$ using natural logarithms ($a = e$):

$\log_9 20 = \frac{\ln 20}{\ln 9} \approx 1.36342$

Exponential and Logarithmic Equations

• Solving Exponential Equations Using the One-to-One Property:
  • Since $f(x) = a^x$ is one-to-one, $a^x = a^y \iff x = y$
  • Example 1(a): Solve $5^x = 125$

$5^x = 5^3 \implies x = 3$

• Example 1(b): Solve $5^{2x} = 5^{x+1}$

$2x = x + 1 \implies x = 1$

• Guidelines for Solving Exponential Equations:

  1. Isolate the exponential expression on one side of the equation.
  2. Take the logarithm of each side, then use the Laws of Logarithms to "bring down the exponent."
  3. Solve for the variable.

• Solving Exponential Equations Examples:

  • Example 2: Solve $3^{x+2} = 7$
  • Taking common logarithm of both sides:

$\log(3^{x+2}) = \log 7$

$(x + 2)\log 3 = \log 7$

$x + 2 = \frac{\log 7}{\log 3}$

$x = \frac{\log 7}{\log 3} - 2 \quad (\text{Exact solution})$

- Calculator approximation: $x \approx -0.228756$

• Example 3: Solve $8e^{2x} = 20$
  • Isolate exponential term by dividing by 8:

$e^{2x} = \frac{20}{8} = 2.5$

$\ln(e^{2x}) = \ln 2.5$

$2x = \ln 2.5$

$x = \frac{\ln 2.5}{2} \approx 0.458$

• Example 4: Solve $e^{3-2x} = 4$
  • Algebraic Solution:

$\ln(e^{3-2x}) = \ln 4$

$3 - 2x = \ln 4$

$-2x = -3 + \ln 4$

$x = \frac{1}{2}(3 - \ln 4) \approx 0.807$

- Graphical Solution: Graph $y = e^{3-2x}$ and $y = 4$. The intersection point gives $x \approx 0.81$.

• Example 5 (Quadratic Type): Solve $e^{2x} - e^x - 6 = 0$
  • Rewrite and factor quadratic in $e^x$:

$(e^x)^2 - e^x - 6 = 0$

$(e^x - 3)(e^x + 2) = 0$

- Zero-Product Property:

$e^x - 3 = 0 \implies e^x = 3 \implies x = \ln 3 \approx 1.0986$

$e^x + 2 = 0 \implies e^x = -2 \quad (\text{No solution because } e^x > 0 \text{ for all } x)$

- The only valid solution is $x = \ln 3 \approx 1.0986$

• Example 6 (Factoring Common Terms): Solve $3xe^x + x^2e^x = 0$
  • Factor out common terms:

$x(3 + x)e^x = 0$

- Divide by $e^x$ (since $e^x \neq 0$ for all real $x$):

$x(3 + x) = 0$

- Solutions: $x = 0$ and $x = -3$

• Solving Logarithmic Equations:
  • Logarithmic functions are one-to-one:

$\log_a x = \log_a y \implies x = y$

• Example 7: Solve $\log_5(x^2 + 1) = \log_5(x - 2) + \log_5(x + 3)$
  • Combine right-hand side using Law 1:

$\log_5(x^2 + 1) = \log_5[(x - 2)(x + 3)]$

$\log_5(x^2 + 1) = \log_5(x^2 + x - 6)$

- Apply one-to-one property:

$x^2 + 1 = x^2 + x - 6$

$1 = x - 6 \implies x = 7$

• Example 8(a): Solve $\ln x = 8$

  • Convert directly to exponential form: $x = e^8 \approx 2981$
  • Alternatively, raise $e$ to both sides: $e^{\ln x} = e^8 \implies x = e^8$

• Example 8(b): Solve $\log_2(25 - x) = 3$

  • Rewrite in exponential form:

$25 - x = 2^3$

$25 - x = 8$

$x = 25 - 8 = 17$

• Example 9: Solve $4 + 3\log(2x) = 16$
  • Isolate logarithmic expression:

$3\log(2x) = 12$

$\log(2x) = 4$

- Convert to exponential form:

$2x = 10^4$

$2x = 10,000$

$x = 5000$