If `4^{2 x}=\frac{1}{32}` then `x` is

Options

A. 5/4
B. 4/5
C. 3/5
D. None of these

Answer

To solve the equation \( 4^{2x} = \frac{1}{32} \), we will follow these steps:

### Step 1: Rewrite the bases
First, we can express both sides of the equation with the same base. We know that \( 4 \) can be rewritten as \( 2^2 \) and \( 32 \) can be rewritten as \( 2^5 \). Thus, we can rewrite the equation as follows:

\[ 4^{2x} = (2^2)^{2x} = 2^{4x} \]

And since \( \frac{1}{32} = 32^{-1} \), we can write:

\[ \frac{1}{32} = \frac{1}{2^5} = 2^{-5} \]

So now our equation looks like this:

\[ 2^{4x} = 2^{-5} \]

### Step 2: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[ 4x = -5 \]

### Step 3: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by \( 4 \):

\[ x = \frac{-5}{4} \]

### Final Answer
Thus, the value of \( x \) is:

\[ x = -\frac{5}{4} \]

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