When `log_(5) x = -2`, what is x?

Options

A. `-32`
B. `-25`
C. `-10`
D. `1/25`

Answer

To solve the equation \( \log_{5} x = -2 \), we can follow these steps:

### Step 1: Understand the logarithmic equation
The equation \( \log_{5} x = -2 \) indicates that \( x \) is the number we get when we raise the base (5) to the power of -2.

### Step 2: Convert the logarithmic equation to exponential form
Using the property of logarithms that states if \( \log_{a} b = c \), then \( b = a^{c} \), we can rewrite the equation:
\[ x = 5^{-2} \]

### Step 3: Calculate \( 5^{-2} \)
Now we need to compute \( 5^{-2} \):
\[ 5^{-2} = \frac{1}{5^2} \]

### Step 4: Simplify \( 5^2 \)
Calculating \( 5^2 \):
\[ 5^2 = 25 \]
Thus,
\[ 5^{-2} = \frac{1}{25} \]

### Step 5: Conclusion
Therefore, the value of \( x \) is:
\[ x = \frac{1}{25} \]

### Final Answer
The value of \( x \) is \( \frac{1}{25} \).

---