If `4^(2x)=(1)/(32)`, then x is

Options

A. `(5)/(4)`
B. `(4)/(5)`
C. `(3)/(5)`
D. `-(5)/(4)`

Answer

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

### Step 1: Rewrite the equation
We start with the equation:
\[ 4^{2x} = \frac{1}{32} \]

### Step 2: Express \( \frac{1}{32} \) as a power
We can rewrite \( \frac{1}{32} \) as \( 32^{-1} \):
\[ 4^{2x} = 32^{-1} \]

### Step 3: Rewrite both sides with the same base
Next, we express both sides with a base of 2. We know that:
\[ 4 = 2^2 \quad \text{and} \quad 32 = 2^5 \]
Thus, we can rewrite the equation:
\[ (2^2)^{2x} = (2^5)^{-1} \]

### Step 4: Simplify the exponents
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we simplify the left side:
\[ 2^{4x} = 2^{-5} \]

### Step 5: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 4x = -5 \]

### Step 6: Solve for \( x \)
Now, we solve for \( x \) by dividing both sides by 4:
\[ x = \frac{-5}{4} \]

Thus, the value of \( x \) is:
\[ \boxed{-\frac{5}{4}} \]

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