https://file.aiquickdraw.com/mathbot/file/1729646288337fvn3ejj5.jpg

The problem given is:

\[
x = \log_5 \left( \frac{1}{25} \right)
\]

To solve this logarithmic equation, we use the properties of logarithms and exponents.

### Step-by-Step Solution:
1. **Rewrite the logarithmic equation in exponential form**:
   \[
   \log_b(a) = c \quad \text{means} \quad b^c = a
   \]
   In our case, \( b = 5 \), \( a = \frac{1}{25} \), and \( c = x \). This gives:
   \[
   5^x = \frac{1}{25}
   \]

2. **Rewrite \( \frac{1}{25} \) as a power of 5**:
   \[
   \frac{1}{25} = 5^{-2}
   \]

3. **Set the exponents equal**:
   Since the bases are the same (both are powers of 5), we can equate the exponents:
   \[
   x = -2
   \]

Thus, the solution is \( x = -2 \).

The solution set is \( \{-2\} \).

Would you like more details, or do you have any questions?

Here are 5 related questions for further practice:
1. Solve \( \log_2 \left( \frac{1}{8} \right) \).
2. Solve \( \log_3 \left( \frac{1}{9} \right) \).
3. Solve \( \log_7 \left( \frac{1}{49} \right) \).
4. Solve \( \log_4 \left( \frac{1}{16} \right) \).
5. Solve \( \log_{10} \left( \frac{1}{1000} \right) \).

**Tip**: Remember, if the logarithm's argument is the reciprocal of a number, the result is the negative of the logarithm of the number!

Solve the logarithmic equation: x = log_5 (1/25)

Learn how to solve the logarithmic equation x = log_5(1/25) step-by-step. This problem covers key concepts of logarithms and exponents, using logarithmic and exponential properties to find x = -2. Suitable for students in Grades 9-11.

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