`log_x(16/25)=-1/2` then `x` is

Options

A. `625/256`
B. `256/625`
C. `526/265`
D. none of these

Answer

To solve the equation \( \log_x\left(\frac{16}{25}\right) = -\frac{1}{2} \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation in exponential form.
The logarithmic equation \( \log_x\left(\frac{16}{25}\right) = -\frac{1}{2} \) can be rewritten in exponential form as:
\[ x^{-\frac{1}{2}} = \frac{16}{25} \]

### Step 2: Simplify the equation.
The expression \( x^{-\frac{1}{2}} \) can be rewritten as:
\[ \frac{1}{\sqrt{x}} = \frac{16}{25} \]

### Step 3: Invert both sides.
To eliminate the fraction, we can take the reciprocal of both sides:
\[ \sqrt{x} = \frac{25}{16} \]

### Step 4: Square both sides to solve for \( x \).
Now, we square both sides to solve for \( x \):
\[ x = \left(\frac{25}{16}\right)^2 \]

### Step 5: Calculate the square.
Calculating the square gives us:
\[ x = \frac{25^2}{16^2} = \frac{625}{256} \]

### Final Answer:
Thus, the value of \( x \) is:
\[ x = \frac{625}{256} \]

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