One answer to the logarithmic equation log5x=logx5 is clearly x=5, but there is a second
solution. How can we find the second solution? Ideas?

To solve the logarithmic equation \( \log_5{x} = \log_x{5} \), we already know that one solution is \( x = 5 \), as the equation is clearly balanced when \( x = 5 \). To find the second solution, we need to explore the equation more deeply.

### Step-by-Step Approach:
1. **Rewrite the equation**: We know that the logarithmic equation \( \log_b{a} \) can be expressed as \( \frac{\log{a}}{\log{b}} \) in terms of common logarithms. So, we rewrite both sides of the equation:
   \[
   \log_5{x} = \frac{\log{x}}{\log{5}}, \quad \log_x{5} = \frac{\log{5}}{\log{x}}
   \]
   Therefore, the equation becomes:
   \[
   \frac{\log{x}}{\log{5}} = \frac{\log{5}}{\log{x}}
   \]

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

3. **Solve for \( \log{x} \)**: Taking the square root of both sides, we get two possible cases:
   \[
   \log{x} = \log{5} \quad \text{or} \quad \log{x} = -\log{5}
   \]

4. **Case 1: \( \log{x} = \log{5} \)**:
   This implies \( x = 5 \), which is the solution we already know.

5. **Case 2: \( \log{x} = -\log{5} \)**:
   This implies \( x = 10^{- \log{5}} = \frac{1}{5} \).

### Conclusion:
The second solution is \( x = \frac{1}{5} \). Thus, the two solutions are:
\[
x = 5 \quad \text{and} \quad x = \frac{1}{5}.
\]

### Tip:
Always remember that when squaring both sides of an equation, you can introduce extraneous solutions, so it's important to check both values back in the original equation to ensure they satisfy the equation.

Would you like further details on any step or have questions about this solution?

Here are 5 related questions to extend your understanding:
1. What are the general properties of logarithmic functions that apply here?
2. How would the equation change if the bases of the logarithms were different?
3. What are other methods to solve equations involving logarithms?
4. How do we handle logarithms with complex numbers?
5. Can logarithmic equations have more than two solutions?

Tip: When dealing with logarithms, always ensure the arguments (inside the logarithms) are positive, as logarithms of non-positive numbers are undefined in the real number system.

One answer to the logarithmic equation log5x = logx5 is clearly x = 5, but there is a second solution. How can we find the second solution? Ideas?

Explore how to solve the logarithmic equation log_5(x) = log_x(5), uncovering both solutions, x = 5 and x = 1/5. This solution applies logarithmic properties, including the change of base formula and solving quadratic-like equations in logarithmic form. Suitable for students in Grades 10-12.

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