Precalculus / Unit 3: Exponential and Logarithmic Functions
3.5 Solving Logarithmic Equations
Notes
- Rewrite in exponential form: $\log_2 (x + 1) = 4 \Rightarrow x + 1 = 2^4$.
- Combine logs with the product or quotient rule first if needed.
- Always check answers: the input of a log must be positive. Reject extraneous solutions.
- Example: $\log x + \log (x - 3) = 1 \Rightarrow x(x - 3) = 10 \Rightarrow x = 5$ or $x = -2$. Only $x = 5$ works.
Practice
Solve $\ln x = 2$.
Exponential form: $x = e^2$.
Solve $\log_5 (3x + 1) = 2$.
$3x + 1 = 25$, so $x = 8$.
Solve $\log x + \log (x - 3) = 1$.
$x^2 - 3x - 10 = 0$ gives $x = 5$ or $-2$, but $-2$ makes a log input negative.
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