The given solution has a misunderstanding due to the incorrect manipulation of logarithmic bases. Let's address each step and identify the mistake:
1. Starting Equation:
[tex]\[
\log x - \log_5 3 = 2 \log_5 3
\][/tex]
Here, [tex]\(\log\)[/tex] represents the common logarithm (base 10), and [tex]\(\log_5\)[/tex] represents the logarithm with base 5. This equation mixes different bases, which complicates direct manipulation.
2. Rewriting the Equation:
Before proceeding directly to solve the equation, let's rewrite it with common bases.
We use the change of base formula [tex]\(\log_5 x = \frac{\log x}{\log 5}\)[/tex] to convert [tex]\(\log_5 3\)[/tex] to base 10 logarithms:
[tex]\[
\log_5 3 = \frac{\log 3}{\log 5}
\][/tex]
3. Substituting into the Equation:
Substituting [tex]\(\log_5 3\)[/tex] into the original equation, we get:
[tex]\[
\log x - \frac{\log 3}{\log 5} = 2 \cdot \frac{\log 3}{\log 5}
\][/tex]
4. Combining Terms:
To combine terms on the left side:
[tex]\[
\log x = \frac{\log 3}{\log 5} + 2 \cdot \frac{\log 3}{\log 5}
\][/tex]
Simplifying the right-hand side:
[tex]\[
\log x = \frac{\log 3}{\log 5} + \frac{2 \log 3}{\log 5}
\][/tex]
[tex]\[
\log x = \frac{\log 3 + 2 \log 3}{\log 5}
\][/tex]
[tex]\[
\log x = \frac{3 \log 3}{\log 5}
\][/tex]
5. Solving for [tex]\(x\)[/tex]:
Converting [tex]\(\log x\)[/tex] from logarithmic form to exponential form:
[tex]\[
x = 10^{\frac{3 \log 3}{\log 5}}
\][/tex]
6. Evaluating the Exponent:
The exponent can be simplified to the form:
[tex]\[
3 \cdot \log_5 3
\][/tex]
To find [tex]\( x \)[/tex], we need to evaluate this using the base 10 logarithms:
[tex]\[
x \approx 111.63968012488048
\][/tex]
So, the corrected solution yields the answer [tex]\( x \approx 111.63968012488048 \)[/tex]. The previous error was the oversight in handling the mixed logarithmic bases.
