if `(25)^(x)=3125` ,then x equals

Options

A. `2//5`
B. `5//2`
C. `1//4`
D. `1//5`

Answer

To solve the equation \( (25)^x = 3125 \), we can follow these steps:

### Step 1: Rewrite the base
We know that \( 25 \) can be expressed as \( 5^2 \). Therefore, we can rewrite the left side of the equation:
\[ (25)^x = (5^2)^x \]

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

### Step 3: Rewrite the right side
Next, we need to express \( 3125 \) as a power of \( 5 \). We can find that:
\[ 3125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5 \]

### Step 4: Set the exponents equal
Now we can rewrite the equation with the same base:
\[ 5^{2x} = 5^5 \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 2x = 5 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by \( 2 \):
\[ x = \frac{5}{2} \]

Thus, the solution is:
\[ x = 2.5 \]