I know it's not the best title but I had no idea how to be specific about it.
Basically what I'm looking for is a rule that states how $$\log^2(a^{f(x)})$$ works. Does it become $$f(x)\log^2(a)$$ or $$f^2(x)\log^2(a)$$?
Thanks for your time :)
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2Does $\log^2 u$ mean $(\log u)^2$, or $\log (\log u)$? – Daniel Fischer May 19 '16 at 19:05
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$\log^2u$ Means $(log u)^2$ – user3501165 May 19 '16 at 19:06
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2Thought so. And now, if you write it $$\Bigl( \log \bigl(a^{f(x)}\bigr)\Bigr)^2,$$ can you see what is correct? – Daniel Fischer May 19 '16 at 19:08
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@DanielFischer yup. Thanks :) – user3501165 May 19 '16 at 19:15
2 Answers
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Since $log^2(a^{f(x)})=log(a^{f(x)})log(a^{f(x)})$. It follows that
$log^2(a^{f(x)})=f(x)log(a)f(x)log(a)=f^2(x)log^2(a)$. Therefore it is the second one.
A.Riesen
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With full parenthesis,
$$\left(\log(a^{f(x)})\right)^2.$$
By a rule of logarithms,
$$\log(a^{f(x)})=f(x)\log(a).$$
Then distributing,
$$\left(f(x)\log(a)\right)^2=f^2(x)\log^2(a).$$