To solve the equation [tex]\( 19 + 2 \ln x = 25 \)[/tex], we will follow these steps:
1. Isolate the logarithmic term:
[tex]\[
19 + 2 \ln x = 25
\][/tex]
Subtract 19 from both sides:
[tex]\[
2 \ln x = 6
\][/tex]
2. Solve for the logarithm:
Divide both sides by 2 to isolate [tex]\( \ln x \)[/tex]:
[tex]\[
\ln x = 3
\][/tex]
3. Exponentiate to eliminate the logarithm:
Recall that [tex]\( \ln \)[/tex] is the natural logarithm, which is the inverse of the exponential function [tex]\( e \)[/tex]. Therefore, we exponentiate both sides with base [tex]\( e \)[/tex]:
[tex]\[
x = e^3
\][/tex]
4. Approximate the value:
Using the known value of [tex]\( e \approx 2.718 \)[/tex], compute:
[tex]\[
e^3 \approx 2.718^3 \approx 20.085
\][/tex]
Thus, the approximate solution to the equation [tex]\( 19 + 2 \ln x = 25 \)[/tex] is:
[tex]\[
x \approx 20.09
\][/tex]
So, the correct answer from the given choices is:
D. [tex]\( x \approx 20.09 \)[/tex]
