Change of Base and the Calculator

A calculator only has two logarithm buttons: log, which means base 10, and ln, which means base e. But you'll constantly need logs in other bases, like log base 7 of 200. The change of base formula rescues you: log base b of v equals log(v) divided by log(b), using any base you want on top and bottom, as long as it's the same base in both places. So log base 7 of 200 becomes log(200)/log(7), two things your calculator can actually compute.

Why does this work? Logs are exponents. log base b of v is the power you raise b to in order to get v. If you call that power y, then b^y = v. Take log (base 10, say) of both sides: y·log(b) = log(v), since log(b^y) = y·log(b) by the power rule. Divide both sides by log(b) and you get y = log(v)/log(b) — exactly the change of base formula. Nothing about this derivation cared that we picked base 10; ln(v)/ln(b) falls out of the identical argument using natural log instead.

This sits right next to the log laws, since the derivation leans on the power rule for logs, and it sits right before solving exponential equations, where you isolate a variable in an exponent and then need to actually evaluate the resulting logarithm as a decimal. Change of base is the bridge between the symbolic log you write down and the number your calculator hands back.