To solve the equation [tex]\(4 \ln x - 8 = 12\)[/tex], follow these steps:
1. Begin by isolating the logarithmic term [tex]\(\ln x\)[/tex]:
[tex]\[
4 \ln x - 8 = 12
\][/tex]
Add 8 to both sides to move the constant term to the right side:
[tex]\[
4 \ln x = 20
\][/tex]
2. Next, we want to isolate [tex]\(\ln x\)[/tex]. To do this, divide both sides of the equation by 4:
[tex]\[
\ln x = 5
\][/tex]
3. To solve for [tex]\(x\)[/tex], we need to remove the natural logarithm. This is done by exponentiating both sides of the equation using the base [tex]\(e\)[/tex] (the base of natural logarithms):
[tex]\[
x = e^5
\][/tex]
4. Calculate [tex]\(e^5\)[/tex]. The value of [tex]\(e^5\)[/tex] is approximately:
[tex]\[
x \approx 148.4131591025766
\][/tex]
5. Now, look at the choices provided:
- A. [tex]\(x \approx 1.61\)[/tex]
- B. [tex]\(x \approx 59,874\)[/tex]
- C. [tex]\(x \approx 10.5\)[/tex]
- D. [tex]\(x \approx 148.4\)[/tex]
6. The choice closest to our calculated value [tex]\( x \approx 148.4131591025766 \)[/tex] is:
[tex]\[
D. x \approx 148.4
\][/tex]
Therefore, the correct answer is [tex]\( \boldsymbol{D. x \approx 148.4} \)[/tex].
