# Math Proof: Natural Log (X ^ Y)

Have you ever wondered why the natural log of X to power of Y equals to Y times the natural log of X? In other words, have you ever wondered why this is true?

**ln (X ^ Y) = Y * ln (X)**

Wonder no more, because here is the proof.

First, we start off with this equation:

**ln (X ^ Y)**

Since X^Y equals to X

_{1}* X

_{2}* .. * X

_{Y}, then we can re-write the starting equation as:

**ln (X**

_{1}* X_{2}* .. * X_{Y})We know that this statement is true:

**ln (A * B) = ln (A) + ln (B)**

Therefore, our equation can be written as:

**ln (X**

_{1}) + ln (X_{2}) + .. + ln (X_{Y})In other words, this is equal to ln (X) exactly Y times or:

**Y * ln (X)**

That’s it! There is your proof for natural log of X to the power of Y is equal to Y times the natural log of X.

**ln (X ^ Y) = Y * ln (X)**

by Phil for Humanity

on 11/04/2009