If `(125)^x = 3125` , then the value of x is

Options

A. `(1)/(5)`
B. `(3)/(5)`
C. `(5)/(3)`
D. `(5)/(7)`

Answer

To solve the equation \( (125)^x = 3125 \), we can follow these steps:

### Step 1: Express both sides with the same base
First, we need to express 125 and 3125 as powers of the same base. We know that:
- \( 125 = 5^3 \)
- \( 3125 = 5^5 \)

### Step 2: Rewrite the equation
Now we can rewrite the original equation using these expressions:
\[ (5^3)^x = 5^5 \]

### Step 3: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify the left side:
\[ 5^{3x} = 5^5 \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 3x = 5 \]

### Step 5: Solve for x
Now, we can solve for \( x \) by dividing both sides by 3:
\[ x = \frac{5}{3} \]

### Final Answer
Thus, the value of \( x \) is \( \frac{5}{3} \).

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