Exponential and Logarithmic Functions: College Algebra Study Guide

Exponential and Logarithmic Functions

Overview

This study guide covers key concepts in exponential and logarithmic functions, including graphing, properties, solving equations, and applications in compound interest. These topics are essential for understanding growth and decay models, as well as for solving equations involving exponents and logarithms in College Algebra.

Graphing Functions and Inverses

• Graphing a Function: To graph a function, plot points that satisfy the equation and connect them smoothly. Check if the function is one-to-one by verifying that each output corresponds to only one input.

• Inverse Functions: Two functions are inverses if their composition yields the identity function: $f(g(x)) = x$ and $g(f(x)) = x$. Use composition to test for inverses.

• Graphing Inverse Functions: The graph of an inverse function is a reflection of the original function across the line $y = x$.

• Example: If $f(x) = 2^x$, its inverse is $f^{-1}(x) = ext{log}_2(x)$.

Exponential Functions

• Definition: An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Graphing: To graph, make a table of values for $x$ and calculate $f(x)$ for at least three points.

• Properties:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • Horizontal asymptote: $y = 0$

• Example: $f(x) = 3^x$

Logarithmic Functions

• Definition: A logarithmic function is the inverse of an exponential function: $y = \log_a(x)$, where $a > 0$, $a \neq 1$.

• Graphing: Make a table of values and plot points. The graph passes through $(1, 0)$ and has a vertical asymptote at $x = 0$.

• Properties:

  • Domain: $(0, \infty)$

  • Range: $(-\infty, \infty)$

  • Vertical asymptote: $x = 0$

• Example: $f(x) = \log_2(x)$

Solving Exponential and Logarithmic Equations

• Exponential Equations: To solve $a^{f(x)} = a^{g(x)}$, set $f(x) = g(x)$ if $a > 0$ and $a \neq 1$.

• Logarithmic Equations: Use properties of logarithms to combine or expand terms, then solve for the variable.

• Example: Solve $2^x = 8$. Since $8 = 2^3$, $x = 3$.

• Example: Solve $\log_3(x) = 2$. $x = 3^2 = 9$.

Properties of Logarithms

• Product Rule: $\log_a(MN) = \log_a(M) + \log_a(N)$

• Quotient Rule: $\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)$

• Power Rule: $\log_a(M^r) = r \log_a(M)$

• Change of Base Formula: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

• Example: $\log_2(8) = \log_2(2^3) = 3 \log_2(2) = 3$

Expanding and Compressing Logarithmic Expressions

• Expanding: Use product, quotient, and power rules to write a single logarithm as a sum or difference of logarithms.

• Compressing: Combine multiple logarithms into a single logarithm using the same rules in reverse.

• Example (Expand): $\log_2(8x) = \log_2(8) + \log_2(x)$

• Example (Compress): $\log_2(8) + \log_2(x) = \log_2(8x)$

Solving Logarithmic and Exponential Equations Using Properties

• Using Common Base: Rewrite both sides of the equation with the same base, then set exponents equal.

• Taking Logarithms of Both Sides: For equations like $a^x = b$, take logarithms: $x = \frac{\log(b)}{\log(a)}$.

• Example: Solve $5^x = 20$. $x = \frac{\log(20)}{\log(5)}$

Applications: Compound Interest and Continuous Compound Interest

• Compound Interest Formula:

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

  • Where $A(t)$ is the amount after $t$ years, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year.

• Continuous Compound Interest Formula:

  • $A(t) = Pe^{rt}$

  • Where $e$ is Euler's number ($\approx 2.718$), $r$ is the annual interest rate, $t$ is time in years.

• Example: If $P = 1000$, $r = 0.05$, $n = 4$, $t = 2$, then $A(2) = 1000 \left[1 + \frac{0.05}{4}\right]^{8}$

Summary Table: Logarithm Properties

Property	Formula	Example
Product Rule	$\log_a(MN) = \log_a(M) + \log_a(N)$	$\log_2(8x) = \log_2(8) + \log_2(x)$
Quotient Rule	$\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)$	$\log_2\left(\frac{8}{x}\right) = \log_2(8) - \log_2(x)$
Power Rule	$\log_a(M^r) = r \log_a(M)$	$\log_2(x^3) = 3 \log_2(x)$
Change of Base	$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$	$\log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)}$

Practice and Examples

• For worked examples, see textbook pages 381-387.

• For practice problems, see pages 388-391 (odd numbers 1-90).

Additional info: This guide expands on the listed topics and formulas, providing definitions, properties, and examples for each concept. The formulas section is based on the "Chapter 5 Formulas" image, which includes key logarithmic and exponential equations relevant to College Algebra.