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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 B.
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