Integrated Math 3 / Unit 3: Exponential and Logarithmic Functions
3.5 Solving Logarithmic Equations
Notes
Isolate the log, then rewrite in exponential form. Combine logs first if needed. Always check that the inputs of every log are positive.
- $\log_3 (x + 1) = 2 \Rightarrow x + 1 = 3^2 \Rightarrow x = 8$.
- $\ln x = 3 \Rightarrow x = e^3$.
- $\log x + \log (x - 3) = 1 \Rightarrow x(x - 3) = 10 \Rightarrow x^2 - 3x - 10 = 0 \Rightarrow x = 5$ or $x = -2$. Only $x = 5$ works, since $\log(-2)$ is undefined.
Practice
Solve $\log_3 (x + 1) = 2$.
$x + 1 = 3^2 = 9$.
Solve $\ln x = 3$.
Rewrite in exponential form with base $e$.
Solve $\log_2 x + \log_2 (x - 2) = 3$.
$x(x - 2) = 8$ gives $(x - 4)(x + 2) = 0$. Since $x = -2$ makes the logs undefined, only $x = 4$ works.
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