Exponential and Logarithmic Functions: Precalculus Chapter 4 Review Study Notes

Exponential and Logarithmic Functions

Introduction

This chapter review covers the fundamental concepts of exponential and logarithmic functions, including their properties, equations, and applications. Mastery of these topics is essential for solving real-world problems involving growth, decay, and compound interest.

Rewriting Between Logarithmic and Exponential Forms

Definition and Conversion

• Exponential Form: An equation of the form $a^x = b$.

• Logarithmic Form: An equation of the form $x = \log_a b$.

• To convert from exponential to logarithmic form: $a^x = b \implies x = \log_a b$.

• To convert from logarithmic to exponential form: $x = \log_a b \implies a^x = b$.

Example: $2^3 = 8$ can be written as $3 = \log_2 8$.

Evaluating Logarithms Without a Calculator

Common Logarithms and Properties

• Logarithm of 1: $\log_a 1 = 0$ for any $a > 0, a \neq 1$.

• Logarithm of the Base: $\log_a a = 1$.

• Power Rule: $\log_a a^k = k$.

• Change of Base: $\log_a b = \dfrac{\log_c b}{\log_c a}$ for any valid base $c$.

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

Expanding and Condensing Logarithmic Expressions

Logarithm Properties

• Product Rule: $\log_a (xy) = \log_a x + \log_a y$

• Quotient Rule: $\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y$

• Power Rule: $\log_a (x^k) = k \log_a x$

Example (Expand): $\log_2 (x^3 y)$ $= \log_2 x^3 + \log_2 y$ $= 3\log_2 x + \log_2 y$

Example (Condense): $2\log_3 x + \log_3 y$ $= \log_3 x^2 + \log_3 y$ $= \log_3 (x^2 y)$

Solving Logarithmic and Exponential Equations

General Strategies

• Isolate the exponential or logarithmic expression.

• Apply logarithms to both sides if the variable is in the exponent.

• Use properties of logarithms to combine or expand expressions as needed.

• Check for extraneous solutions, especially when dealing with logarithms.

Example: Solve $3^x = 7$ Take logarithms: $x \log 3 = \log 7$ $x = \dfrac{\log 7}{\log 3}$

Applications: Compound Interest and Exponential Growth/Decay

Compound Interest

• Formula (compounded n times per year): $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$

• Formula (compounded continuously): $A = Pe^{rt}$

• Variables:

  • $A$ = final amount

  • $P$ = principal (initial amount)

  • $r$ = annual interest rate (decimal)

  • $n$ = number of compounding periods per year

  • $t$ = time in years

Example: If $P = 1000$, $r = 0.05$, $n = 12$, $t = 3$, then $A = 1000\left(1 + \dfrac{0.05}{12}\right)^{12 \times 3}$

Exponential Growth and Decay

• General Formula: $y = y_0 e^{kt}$

• Growth: $k > 0$

• Decay: $k < 0$

Example: Radioactive decay: $y = y_0 e^{-kt}$

Logarithmic Applications: Newton's Law of Cooling

• Formula: $T(t) = T_s + (T_0 - T_s)e^{-kt}$

• $T(t)$ = temperature at time $t$

• $T_s$ = surrounding temperature

• $T_0$ = initial temperature

• $k$ = cooling constant

Example: If $T_0 = 100^\circ$, $T_s = 20^\circ$, $k = 0.1$, $t = 5$, then $T(5) = 20 + (100 - 20)e^{-0.5}$

Half-Life Problems

• Half-life Formula: $y = y_0 \left(\dfrac{1}{2}\right)^{t/h}$

• $h$ = half-life period

• Alternatively, $y = y_0 e^{kt}$ where $k = \dfrac{\ln(1/2)}{h}$

Example: If $y_0 = 120$ grams, $h = 8$ years, $t = 24$ years: $y = 120 \left(\dfrac{1}{2}\right)^{24/8} = 120 \left(\dfrac{1}{2}\right)^3 = 120 \times \dfrac{1}{8} = 15$ grams

Summary Table: Logarithm Properties

Property	Formula	Description
Product	$\log_a (xy) = \log_a x + \log_a y$	Logarithm of a product is the sum of logarithms
Quotient	$\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y$	Logarithm of a quotient is the difference of logarithms
Power	$\log_a (x^k) = k \log_a x$	Logarithm of a power is the exponent times the logarithm
Change of Base	$\log_a b = \dfrac{\log_c b}{\log_c a}$	Convert between different logarithm bases

Additional info: These notes are based on a chapter review worksheet covering exponential and logarithmic functions, including applications to finance and science. All formulas and properties are standard in Precalculus curricula.