Question

solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give the exact answer. then, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution
ln x = 3

rewrite the given equation without logarithms. do not solve for x
x = e^3

solve the equation. what is the exact solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is
(type an exact answer in terms of e.)
b. there are infinitely many solutions.
c. there is no solution.

Explanation:

Step1: Recall the definition of natural logarithm

The natural logarithm $\ln x$ is the logarithm with base $e$, so by the definition of logarithms, if $\ln x = y$, then $x = e^y$.

Step2: Apply the definition to the given equation

Given the equation $\ln x = 3$, using the definition of the natural logarithm (where the base is $e$), we can rewrite it in exponential form. Let $y = 3$ in the formula $x = e^y$. So we get $x = e^3$.

Step3: Determine the solution set

The domain of $\ln x$ is $x>0$. Since $e^3\approx20.09>0$, it is in the domain. And a logarithmic equation of the form $\ln x = k$ (where $k$ is a real number) has exactly one solution. So the solution set is $\{e^3\}$, which means option A is correct.

Answer:

A. The solution set is $\boldsymbol{\{e^3\}}$ (Type an exact answer in terms of $e$)