To find the value of [tex]\( x \)[/tex] in the equation [tex]\( 19 + 2 \ln x = 25 \)[/tex], follow these steps:
1. Isolate the logarithmic term:
[tex]\[
19 + 2 \ln x = 25
\][/tex]
First, subtract 19 from both sides to move 19 to the other side of the equation:
[tex]\[
2 \ln x = 25 - 19
\][/tex]
Simplifying the right side gives:
[tex]\[
2 \ln x = 6
\][/tex]
2. Solve for the natural logarithm:
Divide both sides by 2 to isolate [tex]\( \ln x \)[/tex]:
[tex]\[
\ln x = \frac{6}{2}
\][/tex]
Simplifying gives:
[tex]\[
\ln x = 3
\][/tex]
3. Exponentiate to remove the natural logarithm:
Recall that [tex]\( \ln x \)[/tex] is the natural logarithm, which has the base [tex]\( e \)[/tex]. To solve for [tex]\( x \)[/tex], exponentiate both sides with base [tex]\( e \)[/tex]:
[tex]\[
x = e^3
\][/tex]
4. Calculate the numerical value:
The value of [tex]\( e^3 \)[/tex] is approximately:
[tex]\[
20.085536923187668
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that solves the equation [tex]\( 19 + 2 \ln x = 25 \)[/tex] is closest to:
D. 20.09
