Logarithm Definition

Pre-Calculus · Axiom Academy

A logarithm answers one question: "What exponent turns the base into this number?" The log IS that exponent. 1. The Same Fact, Read Backward You already know 2^3 = 8 . A logarithm just asks the same thing in reverse: " 2 to what power gives 8 ?" The answer is 3 — the exponent — and we write it . Watch the exponent rise to build 8 , then drop back down as the log. Exponential form — the exponent is the input Logarithmic form — the exponent is the answer Strip away the numbers and the rule is one clean equivalence. For a base b , an argument x , and an exponent y : "The log base b of x equals y " means " b raised to the power y gives x ." The two forms carry the same three pieces — only their roles trade places. Notice the exponent y is the same glyph on both sides; the animation lifts it across. The number being raised to a power. The same b sits under the log and as the base of the exponential. The result of the exponentiation — what you feed the log. It's the output b^y , read as the log's input. The exponent in b^y — and the value of the log. This is the punchline: the log is the exponent. The arrow runs both ways: each form is true exactly when the other is. They're two notations for one relationship. Because , a log of a power "cancels" the base and leaves the exponent. That's the whole reason logs and exponentials are called inverse operations — each undoes the other.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.