Solving Logarithmic and Exponential Equations: Precalculus Study Guide

Logarithmic Equations

Introduction to Logarithmic Equations

Logarithmic equations are equations that involve logarithms of algebraic expressions. Solving these equations is a key skill in precalculus, as it prepares students for more advanced topics in calculus and beyond.

• Logarithm Definition: The logarithm base b of a number x is the exponent to which b must be raised to yield x: $\log_b x = y \iff b^y = x$

• Common Bases: Base 10 (common logarithm, $\log$), base e (natural logarithm, $\ln$)

• Solving Strategy: Use logarithm properties to combine or simplify expressions, then solve for the variable.

Properties of Logarithms

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

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

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

• Change of Base Formula: $\log_b x = \frac{\log_a x}{\log_a b}$

Solving Logarithmic Equations: Examples

• Example 1: $\log 5x = \log (2x + 9)$ Solution: Set arguments equal: $5x = 2x + 9 \implies x = 3$

• Example 2: $\log (10 - 4x) = \log (10 - 3x)$ Solution: $10 - 4x = 10 - 3x \implies x = 0$

• Example 3: $\log (4p - 2) = \log (5p + 5)$ Solution: $4p - 2 = 5p + 5 \implies p = -7$

• Example 4: $\log (4k - 5) = \log (2k - 1)$ Solution: $4k - 5 = 2k - 1 \implies k = 2$

Additional info: Always check that solutions do not make the argument of any logarithm zero or negative, as logarithms are undefined for non-positive arguments.

Equations with Multiple Logarithms

• Example 5: $\log (-2a + 9) = \log (7 - 4a)$ Solution: $-2a + 9 = 7 - 4a \implies 2a = 2 \implies a = 1$

• Example 15: $\log x + \log 8 = 2$ Solution: $\log (8x) = 2 \implies 8x = 10^2 = 100 \implies x = 12.5$

• Example 17: $\log 2 + \log x = 1$ Solution: $\log (2x) = 1 \implies 2x = 10^1 = 10 \implies x = 5$

Equations Involving Logarithmic Expressions and Constants

• Example 6: $2\log_x - 2x = 0$ Solution: $2\log x = 2x \implies \log x = x \implies x = 1$ and $x = -2$ (check for valid solutions)

• Example 10: $-6\log_x (x - 3) = -24$ Solution: $6\log_x (x - 3) = 24 \implies \log_x (x - 3) = 4 \implies x - 3 = x^4$ (solve for x)

Exponential and Logarithmic Equations with Natural Logarithms

Introduction to Natural Logarithms

The natural logarithm, denoted $\ln$, is the logarithm with base $e$ (where $e \approx 2.718$). It is commonly used in calculus and higher mathematics.

• Definition: $\ln x = y \iff e^y = x$

• Properties: Similar to other logarithms: $\ln (xy) = \ln x + \ln y$, $\ln (x/y) = \ln x - \ln y$, $\ln (x^k) = k \ln x$

Solving Equations Involving Natural Logarithms

• Example 23: $\ln 2 - \ln (3x + 2) = 1$ Solution: $\ln \left(\frac{2}{3x + 2}\right) = 1 \implies \frac{2}{3x + 2} = e^1 = e \implies 3x + 2 = \frac{2}{e} \implies x = \frac{2}{3e} - \frac{2}{3}$

• Example 24: $\ln (-3x - 1) - \ln 7 = 2$ Solution: $\ln \left(\frac{-3x - 1}{7}\right) = 2 \implies \frac{-3x - 1}{7} = e^2 \implies -3x - 1 = 7e^2 \implies x = -\frac{7e^2 + 1}{3}$

Summary Table: Logarithmic Properties

Property	Formula (LaTeX)	Description
Product Rule	$\log_b (xy) = \log_b x + \log_b y$	Logarithm of a product equals sum of logarithms
Quotient Rule	$\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$	Logarithm of a quotient equals difference of logarithms
Power Rule	$\log_b (x^k) = k \log_b x$	Logarithm of a power equals exponent times logarithm
Change of Base	$\log_b x = \frac{\log_a x}{\log_a b}$	Converts logarithm to a different base

Key Steps for Solving Logarithmic Equations

1. Combine logarithmic terms using properties (product, quotient, power rules).

2. If both sides have a single logarithm with the same base, set the arguments equal.

3. Solve the resulting algebraic equation for the variable.

4. Check all solutions in the original equation to ensure arguments are positive.

Applications

• Exponential Growth and Decay: Logarithmic equations are used to solve for time or rate in growth/decay models.

• pH Calculations: In chemistry, pH is calculated using logarithms: $pH = -\log_{10} [H^+]$

• Sound Intensity: Decibel levels use logarithmic scales.

Additional info: The provided questions and answers cover a wide range of logarithmic and exponential equations typical for precalculus courses, including equations with multiple logarithms, natural logarithms, and applications of logarithmic properties.