Exercise 8.4.7

Question

\begin{enumerate}[label=(\alph*)] 
\item Show $t^{m/n}$ = $(\sqrt[n]{t})^m$ for all $m, n \in \mathbf{N}$.
\item Show $\log(t^x) = x \log t$, for all $t > 0$ and $x \in \mathbf{R}$.
\item Show $t^x$ is differentiable on $\mathbf{R}$ and find the derivative.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item The properties of $e^x$ proved in Exercise 8.4.3 are trivially shown to be true of $t^x$ as well. Then the same strategy as Exercise 8.4.4 can be taken, replacing $e^x$ with $t^x$.
\item Noting that $t^x > 0$, take $\log$ of both sides of the definition of $t^x$.
\item
$$(t^x)' = \left(e^{x \log t}\right)' = \left(e^{x \log t}\right) \log t = t^x \log t$$
 \end{enumerate}

Exercise 8.4.7

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Exercise 8.4.7

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