To solve the equation [tex]\( 2 \ln (2x) + 3 \ln (3x) = 4 \ln (4x) + 2.14 \)[/tex], let's break down the problem step by step.
First, we understand that [tex]\(\ln\)[/tex] denotes the natural logarithm, which is the logarithm to the base [tex]\(e\)[/tex]. Our goal is to find [tex]\(x\)[/tex] such that the equation holds true.
1. Expand the logarithmic terms using the properties of logarithms. Specifically, the property [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex]:
[tex]\[
2 \ln (2x) = 2 (\ln 2 + \ln x) = 2 \ln 2 + 2 \ln x
\][/tex]
[tex]\[
3 \ln (3x) = 3 (\ln 3 + \ln x) = 3 \ln 3 + 3 \ln x
\][/tex]
[tex]\[
4 \ln (4x) = 4 (\ln 4 + \ln x) = 4 \ln 4 + 4 \ln x
\][/tex]
2. Rewrite the original equation substituting the expanded terms:
[tex]\[
2 \ln 2 + 2 \ln x + 3 \ln 3 + 3 \ln x = 4 \ln 4 + 4 \ln x + 2.14
\][/tex]
3. Combine like terms on both sides of the equation where possible:
[tex]\[
(2 \ln 2 + 3 \ln 3) + (2 \ln x + 3 \ln x) = 4 \ln 4 + 4 \ln x + 2.14
\][/tex]
[tex]\[
(2 \ln 2 + 3 \ln 3) + 5 \ln x = 4 \ln 4 + 4 \ln x + 2.14
\][/tex]
4. Isolate the logarithmic terms involving [tex]\(x\)[/tex] on one side by subtracting [tex]\(4 \ln x\)[/tex] from both sides:
[tex]\[
2 \ln 2 + 3 \ln 3 + 5 \ln x - 4 \ln x = 4 \ln 4 + 2.14
\][/tex]
[tex]\[
2 \ln 2 + 3 \ln 3 + \ln x = 4 \ln 4 + 2.14
\][/tex]
5. Isolate [tex]\(\ln x\)[/tex]:
[tex]\[
\ln x = (4 \ln 4 + 2.14) - (2 \ln 2 + 3 \ln 3)
\][/tex]
6. For clarity and precision, compute the values of each logarithm. However, for brevity, we skip numerical simplification steps here and consider the direct numerical results.
Upon solving the equation through algebraic manipulations and using exact logarithm values, we arrive at the solution:
[tex]\[ x \approx 20.15 \][/tex]
To round the solution to the nearest hundredth, we get:
[tex]\[ x \approx 20.15 \][/tex]
Thus, the correct value of [tex]\( x \)[/tex], rounded to the nearest hundredth, is [tex]\( \boxed{20.14} \)[/tex].
