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.
(lnsqrt{x + 7} = 2)
rewrite the given equation without logarithms. do not solve for x.

Explanation:

Step1: Recall the definition of natural logarithm

The natural logarithm $\ln a = b$ is equivalent to $e^b = a$ by the definition of logarithms (where the base of the natural logarithm is $e$).
Given the equation $\ln\sqrt{x + 7}=2$, we can apply this definition. Here, $a=\sqrt{x + 7}$ and $b = 2$.

Step2: Rewrite using the exponential form

Substituting into the exponential form $e^b=a$, we get $e^2=\sqrt{x + 7}$.

Answer:

$e^2=\sqrt{x + 7}$