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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$\begingroup$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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