Answer
True.
The value of x in the function $\ln x-\ln \left( x-8 \right)=1$ is $12.6558$.
Work Step by Step
Consider the given expression.
$\ln x-\ln \left( x-8 \right)=1$ …… (1)
Apply quotient rule for logarithms:
$\begin{align}
& \ln \frac{x}{x-8}=1 \\
& {{\log }_{e}}\left( \frac{x}{x-8} \right)=1 \\
\end{align}$
If any logarithm is in the form of ${{\log }_{b}}M=x$ , then it is equivalent to the $M={{b}^{x}}$
Write the equivalent form of the logarithm.
$\begin{align}
& \frac{x}{x-8}={{e}^{1}} \\
& =2.7183
\end{align}$
Upon further simplification:
$\begin{align}
& x=2.7183x-21.7464 \\
& 1.7183x=21.7464 \\
& x=\frac{21.7464}{1.7183} \\
& =12.6558
\end{align}$
Check:
Substitute $12.6558$ for $x$ in equation (1).
$\begin{align}
\ln \left( 12.6558 \right)-\ln \left( 12.6558-8 \right)\overset{?}{\mathop{=}}\,1 & \\
\ln \left( 12.6558 \right)-\ln \left( 4.6558 \right)\overset{?}{\mathop{=}}\,1 & \\
\ln \left( 12.6558 \right)-\ln \left( 4.6558 \right)\overset{?}{\mathop{=}}\,1 & \\
2.5381-1.5381\overset{?}{\mathop{=}}\,1 & \\
1\overset{?}{\mathop{=}}\,1 & \\
\end{align}$
Thus, it is true.
