Solve for $x$. \[ \ln (x-2)-\ln 5=1 \] Do not round any intermediate computations, and round your answer to the nearest hundredth.

Step 1 ：Given the equation \(\ln (x-2)-\ln 5=1\)

Step 2 ：Using the properties of logarithms, we can combine the two logarithms on the left side of the equation into one. The property we will use is \(\ln a - \ln b = \ln \frac{a}{b}\). So, the equation becomes \(\ln \frac{x-2}{5}=1\)

Step 3 ：We can use the property of logarithms that says \(e^{\ln a} = a\) to get rid of the natural logarithm. So, the equation becomes \(\frac{x-2}{5}=e^1\)

Step 4 ：Solving for x, we get \(x = e^1*5 + 2\)

Step 5 ：Calculating the value of \(x\), we get \(x = 15.591409142295225\)

Step 6 ：Rounding to the nearest hundredth, we get the final answer: \(x = \boxed{15.59}\)