Chapter 4.5, Problem 90E

To determine

Find the value of $k$ .

Answer to Problem 90E

  $\boxed{k=10}$

Explanation of Solution

Given information:

Take logarithms to show that the equation $x^{1/\log x}=5$ has no solution. For what values of $k$ does the equation $x^{1/\log x}=k$ have a solution? What does this tell us about the graph of the function $f\left(x\right)=x^{1/\log x}$ ? Confirm your answer using a graphing device

Calculation:

Graph of the function $f\left(x\right)=x^{1/\log x}$ is,

  

The graph shows that the function is constant and oparallel to $x-axis$ is defined when $x\ge 0$ Consider, $x^{1/\log x}=5$

Taking log both sides,

  $\begin{array}{l}\log x^{1/\log x}=\log 5\\ \frac{1}{\log x}.\log x=\log 5\left[\therefore \log a^b=b\log a\right]\\ 1\times \frac{\log x}{\log x}=\log 5\\ 1\times 1=\log 5\\ 1=\log 5,isimpossible\end{array}$

So, $x^{1/\log x}=5$ has no solution, because $\log 5=0.698\ne 0$

Now ,

  $x^{1/\log x}=k\mathrm{......}\left(1\right)$

Taking log both sides,

  $\begin{array}{l}\log x^{1/\log x}=\log k\\ \frac{1}{\log x}\log x=\log k\\ 1\times 1=\log k\\ 1=\log k\\ {\log }_{10}k=1\left(\therefore \log k={\log }_{10}k\right)\\ k={10}^1\\ k=10\end{array}$

Hence, the solution is, $\boxed{k=10}$