To determine the true solution to the equation [tex]\(2 \ln(4x) = 2 \ln(8)\)[/tex], let's proceed step-by-step.
1. Simplify the Equation:
Start by dividing both sides of the equation by 2:
[tex]\[
\ln(4x) = \ln(8)
\][/tex]
2. Use the Property of Logarithms:
Since the natural logarithm function [tex]\(\ln\)[/tex] is one-to-one, if [tex]\(\ln(a) = \ln(b)\)[/tex], then [tex]\(a = b\)[/tex]. Hence:
[tex]\[
4x = 8
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, solve the equation [tex]\(4x = 8\)[/tex]:
[tex]\[
x = \frac{8}{4}
\][/tex]
[tex]\[
x = 2
\][/tex]
4. Check the Multiple Choice Options:
Among the given choices:
- [tex]\(x = -4\)[/tex]
- [tex]\(x = -2\)[/tex]
- [tex]\(x = 2\)[/tex]
- [tex]\(x = 4\)[/tex]
The correct value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\(x = 2\)[/tex].
Therefore, the true solution to the equation [tex]\(2 \ln(4x) = 2 \ln(8)\)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
