2.4: Logarithms as Inverses | AP Precalculus

How it works

A logarithm answers the exponent question. For a positive base other than one, the logarithm undoes exponentiation. Real logarithms require positive arguments. The inverse exchanges the original function domain and range.

\log_b y=x\quad\Longleftrightarrow\quad b^x=y\quad(b>0,\ b\ne1,\ y>0)

Worked examples

Example 1

Evaluate $\displaystyle \log_2 8$ .

Since $\displaystyle 2^3=8$ , the value is $\displaystyle 3$ .

Example 2

Find the inverse of $\displaystyle f(x)=3^x$ .

Interchange input and output and solve: $\displaystyle f^{-1}(x)=\log_3x$ for $\displaystyle x>0$ .

Problems

Problem 2.4.1

Evaluate $\displaystyle \log_5(1/25)$ .

Show worked solution

Since $\displaystyle 5^{-2}=1/25$ , the logarithm is $\displaystyle -2$ .

Problem 2.4.2

State the domain of $\displaystyle g(x)=\ln(x-4)$ .

Show worked solution

The argument must be positive: $\displaystyle x-4>0$ , so the domain is $\displaystyle (4,\infty)$ .

Problem 2.4.3

Solve $\displaystyle \log_3x=4$ .

Show worked solution

Convert to exponential form: $\displaystyle x=3^4=81$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.4.1

Problem 2.4.2

Problem 2.4.3

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