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.

$ln sqrt{x + 7} = 2$

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 \\(\boxed{}\\). (type an exact answer. type your answer using exponential notation.)
b. there are infinitely many solutions.
c. there is no solution.

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.
a. (type an integer or a decimal rounded to two decimal places as needed.)
b. there are infinitely many solutions.
c. there is no solution.

Explanation:

Step1: Convert logarithmic to exponential

Given \(\ln\sqrt{x + 7}=2\), recall that \(\ln a = b\) is equivalent to \(a = e^b\). So, \(\sqrt{x + 7}=e^2\).

Step2: Square both sides

Square both sides of the equation \(\sqrt{x + 7}=e^2\) to get \(x + 7=(e^2)^2=e^4\).

Step3: Solve for x

Subtract 7 from both sides: \(x = e^4 - 7\).

Step4: Find decimal approximation

Calculate \(e^4\approx54.5982\), then \(x\approx54.5982 - 7 = 47.5982\approx47.60\) (rounded to two decimal places).

Answer:

The exact solution is \(x = e^4 - 7\), and the decimal approximation is \(47.60\). The solution set is a single value, so for the first part (exact solution) the answer is \(e^4 - 7\) (corresponding to option A), and the decimal approximation is \(47.60\).