2.6: Solving Exponential and Logarithmic Equations | AP Precalculus

How it works

Use inverse operations to solve exponential and logarithmic equations. A logarithmic equation over the reals requires every original logarithm argument to be positive. Solving a transformed equation can produce candidates outside that domain, so check the original expressions.

\log_b A=c\quad\Longleftrightarrow\quad A=b^c\quad(A>0)

Worked examples

Example 1

Solve $\displaystyle 2^{x+1}=16$ .

Write $\displaystyle 16=2^4$ . Equal bases give $\displaystyle x+1=4$ , so $\displaystyle x=3$ .

Example 2

Solve $\displaystyle \ln(x-1)=2$ .

Exponentiate: $\displaystyle x-1=e^2$ . Thus $\displaystyle x=1+e^2$ , which satisfies $\displaystyle x>1$ .

Problems

Problem 2.6.1

Solve $\displaystyle 3^{2x}=27$ .

Show worked solution

$\displaystyle 27=3^3$ , so $\displaystyle 2x=3$ and $\displaystyle x=3/2$ .

Problem 2.6.2

Solve $\displaystyle \log_2(x-3)=4$ .

Show worked solution

The domain requires $\displaystyle x>3$ . Rewrite as $\displaystyle x-3=16$ , giving $\displaystyle x=19$ .

Problem 2.6.3

Solve $\displaystyle \ln x+\ln(x-3)=\ln4$ and reject invalid candidates.

Show worked solution

The original domain is $\displaystyle x>3$ . Combine logs to get $\displaystyle x(x-3)=4$ , so $\displaystyle (x-4)(x+1)=0$ . Only $\displaystyle x=4$ remains.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.6.1

Problem 2.6.2

Problem 2.6.3

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