Logarithm Definition

Algebra 2 · Axiom Academy

Understanding logarithms as the inverse of exponential functions — seen through reflection across the line y = x . Start with the exponential function base 2. It takes an input x and raises 2 to that power — hand it an exponent, and it returns a value. For x = 3 , it gives 2^3 = 8 . Feed in an exponent, read off a value 2. What Is an Inverse Function? An inverse function reverses what the original does. If f(a) = b , then the inverse sends b back to a , so f^ -1 (b) = a . Reflecting across y = x turns the point on the exponential into — the x - and y -coordinates simply trade places. The logarithm is the inverse of the exponential. We write it , read “log base 2 of x ,” and it answers exactly one question: what power of 2 equals x ? Valid for a base b > 0 with , and inputs x > 0 . Now reflect not just one point but the entire exponential curve across y = x . Every point mirrors, and the whole curve becomes the logarithm — its inverse. Seeing logarithms as inverses makes them easy to evaluate — just ask “what exponent produces this result?” Every exponential fact has a matching logarithmic one. You've seen the logarithm defined as the inverse of exponentiation — the operation that answers “what power of the base gives this value?” Scroll up to revisit any step.

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