The question a logarithm asks
Exponentiation takes a base and a power and gives a result: 2 raised to the 3rd is 8. A logarithm runs that backwards. It takes the base and the result and gives the power: starting from base 2 and result 8, it asks “to what exponent must I raise 2 to get 8?” The answer is 3. We write this as log₂ 8 = 3, read aloud as “log base two of eight is three.”
So a logarithm is just an exponent in disguise — the exponent you were looking for. The single most useful fact in this whole topic is that the two forms say the same thing, and you can flip freely between them:
EXPONENTIAL FORM <===> LOGARITHM FORM
b^y = x log_b(x) = y
The base b is the same in both.
The log's answer (y) is the exponent.
Examples:
2^3 = 8 becomes log_2(8) = 3
10^2 = 100 becomes log_10(100) = 2
5^0 = 1 becomes log_5(1) = 0
3^(-2) = 1/9 becomes log_3(1/9) = -2Logs and exponentials are inverses
Because each undoes the other, the logarithmic function log_b(x) is the inverse function of the exponential function b^x. The exponential is one-to-one — each output comes from exactly one input — which is precisely the condition that lets it have an inverse at all. Feeding one into the other cancels them: b^(log_b x) = x and log_b(b^x) = x.
Two bases worth memorizing
Two bases are used so often they get shorthand names. The common logarithm is base 10, written simply log x with no subscript — natural for our base-ten number system and for scientific notation. The natural logarithm is base e, written ln x — the inverse of the natural exponential e^x, and the one that appears throughout science. When you see log with no base, assume 10; when you see ln, the base is e.
Common log (base 10), written log: log 1000 = 3 because 10^3 = 1000 log 100 = 2 because 10^2 = 100 log 1 = 0 because 10^0 = 1 log 0.01 = -2 because 10^(-2) = 0.01 Natural log (base e), written ln: ln e = 1 because e^1 = e ln 1 = 0 because e^0 = 1 ln(e^5) = 5 because ln undoes e^(...)