Exponential Functions

Definition and Graphing

Exponential functions are mathematical models where the variable appears in the exponent. The general form is y = a^x, where a is a positive real number not equal to 1.

• Base a: The base a determines the growth or decay rate. If a > 1, the function increases; if 0 < a < 1, it decreases.

• Translations: Shifting the graph horizontally or vertically changes the function to y = a^{x-h} + k.

• Asymptotes: The horizontal asymptote is y = 0 for basic exponential functions.

Example: The graph of y = 2^x passes through (0,1), increases rapidly, and approaches y = 0 as x → -∞.

Applications: Investment Formulas

Exponential functions model compound interest in finance.

• Compounded Continuously:

• Compounded Periodically:

• Variables: P = principal, r = annual rate, n = number of compounding periods per year, t = time in years, A = final amount.

Example: A = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3}$

Logarithmic Functions

Definition and Graphing

Logarithmic functions are the inverse of exponential functions. The general form is y = \log_a(x), where a is the base.

• Domain: The argument of the logarithm must be positive. Set up an inequality: x > 0 for y = \log_a(x).

• Translations: Shifting the graph horizontally or vertically changes the function to y = \log_a(x-h) + k.

• Asymptotes: The vertical asymptote is x = 0.

• Special Logs: log x means log_{10} x (common log); ln x means log_e x (natural log).

Example: The graph of y = \log_2(x) passes through (1,0), increases slowly, and has a vertical asymptote at x = 0.

Evaluating Logarithms

• Definition: if and only if

• Special Values:

Example: because .

Rewriting Logarithms as Exponents

• To rewrite a logarithm as an exponent:

Example: means .

Comparing Logarithmic and Exponential Functions

• One-to-One Property:

  • if and only if

  • if and only if

• Inverse Relationship:

Properties of Logarithms

Expansion Operations

Condensing Operations

Change of Base Formula

Example:

Solving Equations Involving Logs and Exponentials

Solving for t in the Continuously Compounded Interest Formula

To solve for time t in , use logarithms:

• Isolate the exponential:

• Take the natural log:

• Solve for t:

Example: If , , , then

Summary Table: Logarithmic Properties

Test Preparation Tips

• Practice problems from lecture notes and homework.

• Memorize key logarithmic properties and formulas.

• Be able to graph and interpret exponential and logarithmic functions.

• Understand how to solve equations involving exponentials and logarithms, especially in financial contexts.