Let's solve the given equation step-by-step, ensuring we cover each detailed aspect:
Given equation:
[tex]\[
2 \ln\left(e^{\ln(2) \cdot x}\right) - \ln\left(e^{\ln(10) \cdot x}\right) = \ln(30)
\][/tex]
Step 1: Simplify the logarithmic expressions using basic logarithmic properties.
We know that [tex]\(\ln(e^y) = y\)[/tex]. Applying this property:
[tex]\[
\ln\left(e^{\ln(2) \cdot x}\right) = \ln(2) \cdot x
\][/tex]
[tex]\[
\ln\left(e^{\ln(10) \cdot x}\right) = \ln(10) \cdot x
\][/tex]
Substituting these into the equation:
[tex]\[
2 \cdot (\ln(2) \cdot x) - (\ln(10) \cdot x) = \ln(30)
\][/tex]
[tex]\[
2 \ln(2) \cdot x - \ln(10) \cdot x = \ln(30)
\][/tex]
Step 2: Combine like terms.
Factor [tex]\(x\)[/tex] out from the terms on the left side:
[tex]\[
x(2 \ln(2) - \ln(10)) = \ln(30)
\][/tex]
Step 3: Isolate [tex]\(x\)[/tex].
To solve for [tex]\(x\)[/tex], divide both sides by [tex]\(2 \ln(2) - \ln(10)\)[/tex]:
[tex]\[
x = \frac{\ln(30)}{2 \ln(2) - \ln(10)}
\][/tex]
Step 4: Simplify the denominator using properties of logarithms.
We recognize that:
[tex]\[
\ln\left(2^2\right) = 2 \ln(2)
\][/tex]
Then:
[tex]\[
2 \ln(2) - \ln(10) = \ln(4) - \ln(10)
\][/tex]
Using the property of logarithms that [tex]\(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\)[/tex]:
[tex]\[
\ln(4) - \ln(10) = \ln\left(\frac{4}{10}\right) = \ln\left(\frac{2}{5}\right)
\][/tex]
Therefore, the expression for [tex]\(x\)[/tex] becomes:
[tex]\[
x = \frac{\ln(30)}{\ln\left(\frac{2}{5}\right)}
\][/tex]
The expression [tex]\(\frac{\ln(30)}{\ln\left(\frac{2}{5}\right)}\)[/tex] can be further interpreted as [tex]\(\log_{\frac{2}{5}}(30)\)[/tex], but this interpretation steps into advanced logarithmic terms beyond the basics of solving the equation.
The result for [tex]\(x\)[/tex] is therefore:
[tex]\[
x = \log_\frac{2}{5}(30^{\frac{1}{\log(2/5)}})
\][/tex]
Given this, the correct solution to the original equation is not among the provided options (30, 75, 150, 300) directly without a specific transformation, but rather the result we obtained mathematically represents the exact solution:
[tex]\[
x = \log\left(30^{\frac{1}{\log\left(\frac{2}{5}\right)}}\right)
\][/tex]
So, the exact solution to the equation is:
[tex]\[
x = \log\left(30^{\frac{1}{\log(2/5)}}\right)
\][/tex]
