If `log_(100) x=-4`, then the value of x is:

Options

A. `1/(10^(-4))`
B. `(10)^(-4)`
C. `1/(10^(8))`
D. none of these

Answer

To solve the equation \( \log_{100} x = -4 \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation in exponential form
The property of logarithms states that if \( \log_a b = c \), then \( b = a^c \). In this case, we have:
\[ x = 100^{-4} \]

### Step 2: Simplify \( 100^{-4} \)
Next, we need to simplify \( 100^{-4} \). We know that:
\[ 100 = 10^2 \]
Thus, we can rewrite \( 100^{-4} \) as:
\[ 100^{-4} = (10^2)^{-4} \]

### Step 3: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we have:
\[ (10^2)^{-4} = 10^{2 \cdot (-4)} = 10^{-8} \]

### Step 4: Write the final value of \( x \)
Now we can express \( x \) as:
\[ x = 10^{-8} \]

### Conclusion
Thus, the value of \( x \) is:
\[ x = \frac{1}{10^8} \]