change-of-base formula

Most calculators only have two log buttons: common log (base 10) and natural log (base e). So how do you compute a log in some other base, like log base 3 of 20? The change-of-base formula is the bridge: it rewrites any logarithm as a ratio of logs in a base your calculator does know.

The formula is log_b(x) = log_c(x) / log_c(b), where c is any new base you choose (commonly 10 or e). You take the log of the number on top and the log of the original base on the bottom, both in the new base, and divide. Most often people write it as log_b(x) = ln(x) / ln(b) or log(x) / log(b).

It is worth knowing why it works: if y = log_b(x), then b^y = x; taking log base c of both sides gives y · log_c(b) = log_c(x), and dividing isolates y. A practical takeaway is that changing the base only multiplies every value by a fixed constant — logs in different bases are scaled copies of one another, never reshaped.