Logarithmic Expressions

Logarithmic expressions arise naturally as the inverse of exponential expressions. In AP Precalculus, logarithms are used to:

• solve exponential equations

• analyze models involving growth and decay

• manipulate expressions into equivalent, more useful forms

• interpret quantities such as time, magnitude, and scale

Understanding logarithmic expressions is essential for connecting algebraic techniques to real-world applications.

Definition of a Logarithm

A logarithm answers the question:

“To what exponent must the base be raised to produce a given number?”

Formally,

$$y=a^x longrightarrow x=\log_{a}y$$

where:

• a>0, a≠1 (the base)

• y>0 (the argument)

• x is the exponent

Common Logarithmic Bases

Common Logarithm (Base 10)

$$\log{x}=\log_{10}x$$

Frequently used in scientific notation and measurement scales.

Natural Logarithm (Base e)

$$\ln{x}=\log_{e}x$$

Used extensively in growth and decay models, especially continuous processes.

Domain of Logarithmic Expressions

A critical restriction:

The argument of a logarithm must be positive.

Examples:

• $$\log(x-3)$$ requires $$x-3 > 0 \longrightarrow x > 3$$

• $$\ln(2x+1)$$​ requires $$2x+1 > 0 \longrightarrow x > -\frac{1}{2}$$

Failure to apply domain restrictions is a common AP exam error.

Laws of Logarithms

Logarithmic manipulation relies on three core laws:

Product Rule

$$\log_{a}{xy}=\log_{a}x+\log_{a}y$$

Quotient Rule

$$\log_{a}\left(\frac{x}{y}\right)=\log_{a}x-\log_{a}y$$

Power Rule

$$\log_{a}(b^x)=x\cdot\log_{a}b$$

These rules allow complex expressions to be rewritten and simplified.

Expanding Logarithmic Expressions

Example:

$$\log(5x^2y)$$

Step-by-step:

$$\log5+\logx^2+\log{y}$$

Apply power rule:

$$\log5+2\logx+\log{y}$$

Condensing Logarithmic Expressions

Example:

$$3\log{x}-\frac{1}{2}\log{y}+\log4$$

Rewrite using laws:

$$\log{x^3}-\log{y^{\frac{1}{2}}}+\log4$$

Combine:

$$\log\left(\frac{4x^3}{\sqrt{y}}\right)$$

Change of Base Formula

When a calculator does not support a certain base:

$$\log_{a}x=\frac{\log{x}}{\log{a}}=\frac{\ln{x}}{\ln{a}}$$​

This allows evaluation and comparison of logarithmic expressions.​

Graphical Interpretation

Key features of $$f(x)=\log_{a}x$$:

• Domain: (0,∞)

• Vertical asymptote: x=0

• x-intercept: (1,0)

• Increasing if a>1

• Decreasing if 0<a<1

Summary

Logarithmic expressions:

• are inverses of exponential expressions

• require strict domain considerations

• follow specific algebraic laws

• enable solution of exponential equations

Mastery of logarithmic expressions is essential for success in AP Precalculus and provides a foundation for calculus, statistics, and applied mathematics.