Study Notes on Exponential and Logarithmic Functions

PreCalculus: Chapter 3 Section 1 - Exponential and Logarithmic Functions

Breeding Potential for Deer Population

• Model Assumptions:

  • No deaths due to accident, disease, hunting, or predators.

  • Over a span of 7 years, a single breeding pair can produce a population exceeding 40 deer given optimal conditions.

Operations of Exponents

• Exercises on exponent operations:

  • $3^5 = ?$

  • Why? (Reasoning for operation)

  • $5^2 = ?$

  • Why? (Reasoning for operation)

  • $(2)^3 = ?$

  • Why? (Reasoning for operation)

Algebraic and Transcendental Functions

• Algebraic Functions:

  • Defined as functions utilizing algebraic operations: addition, multiplication, subtraction, and division.

• Transcendental Functions:

  • Functions whose operations transcend traditional algebraic operations.

  • Examples include exponential and logarithmic functions as well as trigonometric functions.

  • They do not conform to the same rules of addition, subtraction, multiplication, and division established for algebraic functions.

Identifying Exponential Functions

• Examples of Exponential Functions:

  • $f(x) = 3^x$ - Yes, this is an exponential function.

  • $f(x) = 3^{-x}$ - Yes, this is also an exponential function.

  • $f(x) = 3$ - No, this does not qualify as an exponential function since the exponent is not a variable (this is a power function).

Exponential Functions Defined

• General Form:

  • $y = ab^x$

  • Where:

    • $x$: Any real number (variable)

    • $a$: Real number constant

    • $b$: Real number constant ($b > 0$, $b <br>eq 1$)

• Characteristics:

  • The defining characteristic of an exponential function is that the power is the variable

Exponential Growth

• Expression for Growth:

  • $f(x) = ab^x$ where $b > 1$.

• Questions for Analysis:

  • Y-intercept

  • X-intercept

  • Domain

  • Range

  • Asymptote

  • End Behavior

  • Continuity

Exponential Decay

• Expression for Decay:

  • $f(x) = ab^x$ where $0 < b < 1$ (indicates decay with fractions, decimals, or negative exponents).

• Questions for Analysis:

  • Y-intercept

  • X-intercept

  • Domain

  • Range

  • Asymptote

  • End Behavior

  • Continuity

Graphing Exponential Functions

• Graph Transformations:

  • Original Function: $f(x) = 2^x$

  • Compared to transformations:

  • $f(x) = 2^{-x} ext{ (reflection)}$

  • $f(x) = -2^x ext{ (reflection across the x-axis)}$

  • $f(x) = 2^{x-1} ext{ (horizontal shift)}$

The Exponential Family of Functions

• General Form:

  • $f(x) = -h + k$

  • Transformed Characteristics:

    • $h$: Horizontal shift (opposite direction of the sign)

    • Negative exponent influences decay curve.

    • $b$: Affects the steepness of the curve and where it crosses the y-axis.

    • $k$: Adjusts the vertical shift, moving upward/downward based on its sign.

One-to-One Property of Exponential Functions

• Example:

  • For the equation: $3^{2+5} = 27$

  • When expressed with a common base, the exponents must be equal:

  • $3^{2+5} = 3^{3}$

  • Thus, $2 + 5 = 3$.

Solving Exponential Equations

• Try It: Solve the following:

  • $25^3 = 125^6$

  • Simplifying:

  • $5^3 = 5^6$

  • $6 = 18$

  • Which results in $3$

Population Growth Example

• Initial Forest Growth: A 500 acre forest grows at an average rate of 3% annually.

• Yearly Calculation Sequence:

  • Year 1: $500 + 500 imes 0.03 = 515$

  • Year 2: $515 + 515 imes 0.03 = 530.45$

  • Year 3: $530.45 + 530.45 imes 0.03 = 546.36$

  • Year 4: $546.36 + 546.36 imes 0.03 = 562.75$

• Efficient Calculation Method:

  • $500(1.03)^{12} = 712.88$

Exponential Growth & Decay Formulas

• General Relationships:

  • For growth, the formula is:

  • $N = N_0(1 + r)^t$

    • If $r > 0$ it indicates growth.

    • If $r < 0$ it indicates decay.

    • $N_0$: Initial amount, $r$: Rate, $t$: Time.

Compound Interest Formula

• General Formula:

  • $A = P(1 + rac{r}{n})^{nt}$

  • Where:

    • $A$: Amount after time t

    • $P$: Principal amount (initial investment)

    • $n$: Number of times the interest is compounded in a year

    • $t$: Number of years

    • $r$: Interest rate as a decimal.

Example of Compound Interest Calculation

• Scenario: Investment of $20,000 for 3 years at 6% compounded daily.

• Formula Application:

  • $A = 20000(1 + rac{0.06}{365})^{365 imes 3}$

  • Result: $A ≈ 23943.99$

Effects of Compounding Frequency

• Comparison of Amounts Based on Compounding Frequency:

  • Invest $100 at 5% for 1 year:

  • Compounded Annually: $105.00$

  • Compounded Semiannually: $105.06$

  • Compounded Quarterly: $105.09$

  • Compounded Monthly: $105.12$

  • Compounded Daily: $105.13$

  • Compounded Hourly: $105.13$

Continuous Compounding Formula

• Formula:

  • $A = Pe^{rt}$

  • Natural Base:

  • Form derived from: $ext{lim}_{n o ext{∞}} (1 + rac{1}{n})^n <br>ightarrow e ≈ 2.7182…$

  • This base (e) is preferred in real-world applications of exponential growth.

Carbon-14 Dating Example

• Decay Rate:

  • Carbon-14 isotope decay rate: -0.012%.

• Initial Amount:

  • If a bone contains 3.4 grams of Carbon-14, the amount remaining after 450 years is calculated as:

  • $N = 3.4(1 - 0.00012)^{450} <br>ightarrow N ≈ 3.22 ext{ grams}$

Half-Life Calculation Example

• Radium Decay:

  • Half-life: 1620 years.

• Calculation:

  • Starting mass: 10 lbs.

  • Quantity left after 850 years:

  • $N = 10(1/2)^{ rac{850}{1620}} <br>ightarrow N ≈ 6.95 ext{ lbs.}$

Population Prediction Example

• Initial Population: 7,500, with a continuous growth rate of 2.3%.

• Population after 8 years calculation:

  • $N = 7500e^{0.023 imes 8} <br>ightarrow N ≈ 9015$

Turkey Consumption Projection Example

• Initial Year: t = 0 corresponds to 1939.

• Model for Consumption:

  • $f(t) = 2.3(3)^{0.033t}$

• Prediction for Year 2010:

  • $f(71) ≈ 30.2 ext{ lbs.}$

Distinction Between Growth and Decay in Exponential Functions

• For Exponential (not continuous):

  • $N = N_0(1 + r)^t$

• For Continuous Exponential:

  • $N = N_0e^{rt}$

  • Where $r > 0$ indicates growth, and $r < 0$ indicates decay.

Exponential Rate of Increase Example

• Initial Data: 125 deer in 2000, increasing to 264 deer in 2010.

• Modeling Method:

  • Use the equation $N(t) = N_0(1 + r)^t$

  • Find $r$ in the context of data provided for accurate growth modeling.

Another Turkey Consumption Example

• Growth Model Established: Similar to previous example with different initial conditions leading to:

  • $f(t) = 2.3(3)^{0.033t}$

• Amount consumed in 2010:

  • Resulting in $f(t) = 73$.

Assignment Instructions

• Complete exercises P-226 #3-21 odd, 29-33 odd, 47-67 odd.

Final Notes on Operations and Graphs of Exponentials

• Additional exercises answering questions regarding transformations and comparisons of different functions accurate to properties of exponential graphs.