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\
ewline rewrite the given equation without logarithms. do not solve for x.\
ewline 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.\
ewline a. the solution set is \\{e^3\\}. (type an exact answer in terms of e.)\
ewline b. there are infinitely many solutions.\
ewline c. there is no solution.\
ewline what is the decimal approximation for the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\
ewline a. the solution is \underline{}. (type an integer or a decimal rounded to two decimal places.)\
ewline b. there are infinitely many solutions.\
ewline c. there is no solution.

Explanation:

Step1: Recall logarithm definition

The equation is $\ln x = 3$. By the definition of the natural logarithm, if $\ln a = b$, then $a = e^b$.

Step2: Apply the definition

Applying this to $\ln x = 3$, we get $x = e^3$. This is the exact solution.

Step3: Find decimal approximation

Calculate $e^3$. We know that $e\approx2.71828$, so $e^3\approx2.71828^3\approx20.09$.

Answer:

Exact solution: $e^3$; Decimal approximation: $20.09$