ACT Preparation / Unit 12 ACT 1120 worked example, 60 to 90 seconds
Solving a Logarithmic Equation
The idea
A logarithmic equation is solved in four steps. First, use the log rules to combine the logs on each side into a single log. Second, if one log equals a number, rewrite in exponential form; if one log equals another log of the same base, set their inputs equal. Third, solve. Fourth, check every answer: each log's input in the original equation must be positive.
In practice
Worked example: log₂ x + log₂ (x − 2) = 3. Step one: log₂ (x(x − 2)) = 3. Step two: x(x − 2) = 2³ = 8. Step three: x² − 2x − 8 = 0, so (x − 4)(x + 2) = 0, and x = 4 or x = −2. Step four: x = −2 makes log₂ x undefined, so the only solution is x = 4.
Where people slip
Combining the logs hides the original domain, which is why the check is required: x = −2 solves the combined equation but not the original.
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