ACT Preparation / Unit 12 ACT 1112 worked example, 60 to 90 seconds
Evaluating Logarithms
The idea
To evaluate a logarithm without a calculator, work in three steps. First, set it equal to y and rewrite in exponential form: log_b(x) = y becomes bʸ = x. Second, write x as a power of b, using negative exponents for fractions and fractional exponents for roots. Third, read y from the matching exponents.
In practice
Worked example: log₃ 81: 3ʸ = 81 = 3⁴, so y = 4. log₂ (1/16): 2ʸ = 2⁻⁴, so y = −4. log₄ 2: 4ʸ = 2, and 2 = 4^(1/2), so y = 1/2. log₉ 27: 9ʸ = 27; both are powers of 3, so 3^(2y) = 3³, and y = 3/2.
Where people slip
When the base and the input are not powers of each other, look for a common base, as with 9 and 27. If there is none, the change-of-base formula gives a decimal.
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