Definition: The change of base formula rewrites a logarithm in any base as a quotient of logarithms in a new base: log base b of x equals log x divided by log b.
What does the formula say?
For positive b not equal to 1 and any positive base c,
logb(x) = logc(x) / logc(b).
Calculators usually have only the common log (base 10) and the natural log, so you pick c = 10 or c = e to evaluate a logarithm in an unusual base.
How do you apply it?
Worked example: evaluate log7(100). Use the natural log: ln 100 / ln 7 = 4.6052 / 1.9459, which is about 2.37.
Check: 72 = 49 and 73 = 343, so an answer between 2 and 3 makes sense for 100.
The formula also solves equations such as 3x = 20. Rewrite as x = log3(20) = ln 20 / ln 3, about 2.73.
What mistake happens most?
Flipping the fraction. The base you are leaving goes in the denominator. Another trap is cancelling the “log” as if it were a factor, writing log x / log b as x / b. Logs are functions, not multipliers.