Solving Log Equations

Method 1 — Convert to exponential form:
$\log_b(x) = c \quad \Rightarrow \quad x = b^c$

$log₃(x) = 4$
$x = 3⁴$ convert to exponential
$x = 81$

Solving log₃(x) = 4 by converting to exponential form

Method 2 — If both sides are logs with the same base:
$\log_b(A) = \log_b(B) \quad \Rightarrow \quad A = B$

Method 3 — Combine logs, then convert:
Use log properties to combine into a single log, then convert.

Always check: The argument of a log must be positive. Reject any solution that makes the argument $\leq$ 0.

Example 1

Solve $\log_2(x + 3) = 5$

Convert: $x + 3 = 2^5 = 32$

$x = 29$

Check: $\log_2(32) = 5$ ✓

x = 29

Example 2

Solve $\log(x) + \log(x - 3) = 1$

Combine: $\log(x(x-3)) = 1$

Convert ( $\log$ = $\log_{10}$ ): $x(x-3) = 10$

$x^2 - 3x - 10 = 0$

$(x-5)(x+2) = 0$

$x = 5$ or $x = -2$

Check: $x = -2$ makes $\log(-2)$ undefined. Reject.

$x = 5$ ✓

x = 5

Tip

ACT Pro Tip

Most ACT log equations only require Method 1 (convert to exponential). If you see $\log_b(\text{expression}) = \text{number}$ , just rewrite as $\text{expression} = b^{\text{number}}$ and solve the resulting algebra.

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