Calculate the value of x from log(324)/log(18) = log(x).

Concept

For a logarithm ratio log(a)/log(b), this equals logb(a) — the ratio's value is independent of which base is used for log itself, as long as the SAME base is used top and bottom. Also, the power rule states log(an) = n·log(a) for any base. Finally, by the definition of a logarithm, log(x) = k means x = (base)k.

Application

1. Recognize that 324 can be expressed as a power of 18: 18 × 18 = 324, i.e. 324 = 182.

2. Rewrite the numerator using the power rule: log(324) = log(182) = 2·log(18).

3. Substitute back into the ratio: log(324)/log(18) = 2·log(18)/log(18) = 2.

4. The original equation becomes log(x) = 2.

5. By the definition of a (base-10) logarithm, log(x) = 2 means x = 102 = 100.

Cross-check

Substituting x = 100 back: log(100) = log(102) = 2, which matches log(324)/log(18) = 2 computed above (numerically, log(324) ≈ 2.5105 and log(18) ≈ 1.2553, and their ratio ≈ 2). This confirms x = 100.