To solve the equation [tex]\(4 \ln x - 8 = 12\)[/tex], follow these steps:
1. Isolate the logarithmic term:
[tex]\[
4 \ln x - 8 = 12
\][/tex]
Add 8 to both sides to isolate the logarithmic expression:
[tex]\[
4 \ln x = 20
\][/tex]
2. Divide by 4 to solve for [tex]\(\ln x\)[/tex]:
[tex]\[
\ln x = \frac{20}{4}
\][/tex]
[tex]\[
\ln x = 5
\][/tex]
3. Exponentiate both sides to solve for [tex]\(x\)[/tex]:
The natural logarithm function [tex]\(\ln\)[/tex] is the inverse of the exponential function with base [tex]\(e\)[/tex]. Therefore, to undo the logarithm, exponentiate both sides with base [tex]\(e\)[/tex]:
[tex]\[
x = e^5
\][/tex]
4. Compute the numerical value:
Using a calculator or a tables of exponents, we find:
[tex]\[
e^5 \approx 148.413159
\][/tex]
Thus, the approximate value of [tex]\(x\)[/tex] is 148.413159. Comparing this to the given choices, the closest approximation is [tex]\(x \approx 148.4\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{x \approx 148.4}
\][/tex]
