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I understand that the power rule for logarithms indicate that $\log(x^y)=y\cdot\log(x)$, but how about for $\log(x^{y^z})$? Does this equal $y^z\log(x)$ or $yz\cdot\log(x)$ to follow the power rule?

Explanation for both viewpoints:

The option $y^z\log(x)$ seems logical because following the power rule $\log(x^y)=y\cdot\log(x)$, simply bring out the exponent of x.

The option $yz\cdot\log(x)$ seems logical because following the power rule, bring out each exponent.

Which is the correct option?

Victor Resnov
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  • let $v = y^z$ so that $\log(x^v)=v\log(x)$, now what happens ? – wd violet Aug 26 '21 at 15:07
  • The first one is correct. The general rule is $\log\left(A^B \right)= B \log A$, and you are applying it to the case $B = y^z, A = x$. In your second attempt, you say you are using the same rule. In that case, for what $A$ and $B$ are you applying the rule? – Anonymous Aug 26 '21 at 15:08
  • Well, generally it's not true that $x^{y^z}$ and $\left(x^y\right)^z$ are equal. – peterwhy Aug 26 '21 at 15:09
  • It is a convention that when we write $x^{y^z}$, what we mean is $x^{(y^z)}$, not $\left(x^y\right)^z$. If you are misinterpreting $x^{y^z}$ to mean the second thing, then it is to be expected that you will find $yz \log x$ for its logarithm. – Anonymous Aug 26 '21 at 15:11

2 Answers2

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The correct option is the first one. Assuming that $x>0$ and that $a\in\Bbb R$, you always have $\log\left(x^a\right)=a\log(x)$. So, when $a=y^z$, you have\begin{align}\log\left(x^{y^z}\right)&=\log\left(x^a\right)\\&=a\log(x)\\&=y^z\log(x).\end{align}The “bring out every exponent” rule is not really a rule, at least not in the sense that you apply it to literally every level of the expression that you are dealing with.

Joe
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José Carlos Santos
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Note that $y^z$ is one number. To avoid confusion you can let it equal to $a$

Therefore $ \displaystyle\log x^{(y^{z})}=\log x^a=a\log x=y^z\log x$

Also $yz\log x=\log x^{yz}\neq \log x^{y^{z}}$

Lalit Tolani
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