Exponential and Logarithmic Functions: Study Guide (Sections 4.1–4.4)

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: $A = P e^{rt}$

• Compounded Periodically: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

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

Example: $1000 invested at 5% compounded quarterly for 3 years:$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: $\log_a x = y$ if and only if $a^y = x$

• Special Values:

  • $\log_a 1 = 0$

  • $\log_a a = 1$

  • $\log x = \log_{10} x$

  • $\ln x = \log_e x$

Example: $\log_3 9 = 2$ because $3^2 = 9$.

Rewriting Logarithms as Exponents

• To rewrite a logarithm as an exponent: $\log_a x = y \Rightarrow a^y = x$

Example: $\log_5 25 = 2$ means $5^2 = 25$.

Comparing Logarithmic and Exponential Functions

• One-to-One Property:

  • $a^u = a^v$ if and only if $u = v$

  • $\log_a u = \log_a v$ if and only if $u = v$

• Inverse Relationship:

  • $\log_a a^u = u$

  • $a^{\log_a u} = u$

Properties of Logarithms

Expansion Operations

• $\log_a(MN) = \log_a M + \log_a N$

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

• $\log_a(M^p) = p \log_a M$

Condensing Operations

• $\log_a M + \log_a N = \log_a(MN)$

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

• $p \log_a M = \log_a(M^p)$

Change of Base Formula

• $\log_a x = \frac{\log x}{\log a} = \frac{\ln x}{\ln a}$

Example: $\log_2 8 = \frac{\ln 8}{\ln 2} = 3$

Solving Equations Involving Logs and Exponentials

Solving for t in the Continuously Compounded Interest Formula

To solve for time t in $A = P e^{rt}$, use logarithms:

• Isolate the exponential: $\frac{A}{P} = e^{rt}$

• Take the natural log: $\ln\left(\frac{A}{P}\right) = rt$

• Solve for t: $t = \frac{\ln\left(\frac{A}{P}\right)}{r}$

Example: If $A = 2000$, $P = 1000$, $r = 0.05$, then $t = \frac{\ln(2)}{0.05}$

Summary Table: Logarithmic Properties

Property	Formula	Example
Product Rule	$\log_a(MN) = \log_a M + \log_a N$	$\log_2(8 \times 4) = \log_2 8 + \log_2 4$
Quotient Rule	$\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N$	$\log_3\left(\frac{27}{3}\right) = \log_3 27 - \log_3 3$
Power Rule	$\log_a(M^p) = p \log_a M$	$\log_5(25^2) = 2 \log_5 25$
Change of Base	$\log_a x = \frac{\ln x}{\ln a}$	$\log_2 8 = \frac{\ln 8}{\ln 2}$

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.