Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give an exact answer. then, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.
\\(\ln \sqrt{x + 9} = 2\\)
rewrite the given equation without logarithms. do not solve for x

Explanation:

Step1: Recall the definition of natural logarithm

The natural logarithm $\ln y$ is the inverse of the exponential function $e^x$. That is, if $\ln y = a$, then $y = e^a$.
Here, we have $\ln\sqrt{x + 9}=2$, so we can rewrite this using the exponential form. Let $y=\sqrt{x + 9}$ and $a = 2$. Then by the definition of natural logarithm, we get $\sqrt{x + 9}=e^2$.

Step2: Square both sides (optional, but to get rid of the square root for further solving if needed)

If we want to rewrite the equation without the square root (though the problem says not to solve for $x$, but just to rewrite without logarithms), we can square both sides of $\sqrt{x + 9}=e^2$.
Squaring both sides, we have $(\sqrt{x + 9})^2=(e^2)^2$.
Simplifying, we get $x + 9 = e^{4}$. But the main step to rewrite without logarithms is from $\ln\sqrt{x + 9}=2$ to $\sqrt{x + 9}=e^2$.

Answer:

$\sqrt{x + 9}=e^2$ (or $x + 9 = e^{4}$ if we square both sides, but the key rewrite without logarithm is using the exponential form of the natural log)