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The derivative of $ (\log_2 n)^5$? (log base 2) Hey everyone, I am not sure how to go about this question because I am not sure what to do with the power $5$ ( or any other power ) in the log? Should I try to convert it to a natural log ? How ? Any help ? (Note: this is not a HW problem, I just want to learn how to solve such a problem where the log is raised to a power) Thanks in advance

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    Trying to convert it to a natural log is always a good idea, because the derivative of those are easy. Do you know how to derive $(\log_e(x))^5$? Do you know what the chain rule is? – 2017-02-24
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    Do you mean $\log_{2}^{5}(n)$? Does $n$ denote a real variable? – 2017-02-24
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    I know the chain rule, and I know that to convert log n ( base 2 ) to a natural log it would be ln n / ln 2. But how about the exponential? ^5? I am not sure how to deal with it in the conversion. And no I mean (log2n)^5. – 2017-02-24

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Note that I'm implicitly assuming that $n \in \mathbb R$. First, we observe that

$$\log_2 n=\log(n)/\log(2)$$ where $\log$ is the natural logarithm. Hence,

$$(\log_2n)^5=\left(\frac{\log n}{\log 2}\right)^5=\frac{1}{\log(2)^5} \cdot \log(n)^5.$$

To find the derivative, we use the chain rule: $g(f(x))^\prime=g^\prime(f(x))\cdot f^{\prime}(x)$.

Note that $g=(\log_2n)^5=\frac{1}{\log(2)^5} \cdot x^5$ and $f=(\log(n))$ here.

Can you find these derivatives seperately and use the chain rule to finish?

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    @Rain Another possibility is to take $g(x)=x^5$ and $f(x)=\frac{\ln(x)}{\ln(2)}$ – 2017-02-24