Chapter 1, Problem 10RQ

To determine

Whether the statement, if $x>0$, then ${\left(\ln x\right)}^6=6\ln x$ is true or false.

Answer to Problem 10RQ

Solution:

The statement is false.

Explanation of Solution

Recall the logarithm rule, $\ln x^m=m\ln x$.

Thus, $\ln x^6=6\ln x$.

Since ${\left(\ln x\right)}^6$ and $\ln x^6$ are not the same, ${\left(\ln x\right)}^6\ne 6\ln x$.

For example, let x = 3 and compute the value of ${\left(\ln x\right)}^6$ and $\ln x^6$.

$\begin{array}{c}{\left(\ln 3\right)}^6={\left(1.098\right)}^6\\ =1.752\\ \ln 3^6=\ln 729\\ =6.591\end{array}$

Thus, ${\left(\ln x\right)}^6\ne 6\ln x$.

Therefore, the given statement is false.