https://file.aiquickdraw.com/mathbot/file/1725609921350bn7ftyhu.jpg

The equation in the image is:

\[
\log(x) = \log_5(2x)
\]

### Step-by-step solution:

1. **Rewrite the logarithms using their properties**:
   The equation has two different bases for logarithms: base 10 for \(\log(x)\) and base 5 for \(\log_5(2x)\).

   We can use the change of base formula to rewrite \(\log_5(2x)\) in terms of \(\log(x)\):
   \[
   \log_5(2x) = \frac{\log(2x)}{\log(5)}
   \]

   So the equation becomes:
   \[
   \log(x) = \frac{\log(2x)}{\log(5)}
   \]

2. **Multiply both sides by \(\log(5)\)** to eliminate the denominator:
   \[
   \log(5) \cdot \log(x) = \log(2x)
   \]

3. **Expand the logarithm on the right-hand side**:
   Use the property \(\log(ab) = \log(a) + \log(b)\) for \(\log(2x)\):
   \[
   \log(5) \cdot \log(x) = \log(2) + \log(x)
   \]

4. **Move all terms involving \(\log(x)\) to one side**:
   Subtract \(\log(x)\) from both sides:
   \[
   \log(5) \cdot \log(x) - \log(x) = \log(2)
   \]
   Factor out \(\log(x)\) on the left:
   \[
   \log(x) \cdot (\log(5) - 1) = \log(2)
   \]

5. **Solve for \(\log(x)\)**:
   \[
   \log(x) = \frac{\log(2)}{\log(5) - 1}
   \]

6. **Solve for \(x\)**:
   Since \(\log(x)\) is the logarithm to base 10, convert back to the exponential form:
   \[
   x = 10^{\frac{\log(2)}{\log(5) - 1}}
   \]

This is the exact solution for \(x\). If you'd like, I can compute the numerical value of \(x\).

Would you like a more detailed explanation or a numeric solution? 

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### Related questions:
1. What are the properties of logarithms, such as product and quotient rules?
2. How does the change of base formula work?
3. What is the difference between logarithms with different bases?
4. How can you solve logarithmic equations involving multiple bases?
5. Can this equation have more than one solution?

**Tip:** Understanding the properties of logarithms is key to solving equations that involve them efficiently.

Learn how to solve the logarithmic equation log(x) = log_5(2x) using the change of base formula and properties of logarithms. This problem covers key concepts like exponential equations and logarithmic rules, suitable for students in Grades 9-12.

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