Phil for Humanity Phil for Humanity
A Guide for the Survival of Humankind and Helping the World, Society, and Yourself.

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 X1 * X2 * .. * XY, then we can re-write the starting equation as:

ln (X1 * X2 * .. * XY)

We know that this statement is true:

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

Therefore, our equation can be written as:

ln (X1) + ln (X2) + .. + ln (XY)

In other words, this is equal to ln (X) exactly Y times or:

Y * ln (X)

Thatís 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

Related Articles
 » Proof: 0 * Undefined = -1
 » What does Zero Times Infinity Equal?
 » Undefined * Undefined = Undefined