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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)


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

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