Solve Exponential Equation: (4/7)^x = 2401/256

Published on March 7, 2026

Algebra

Solve for x in the exponential equation (4/7)^x = 2401/256 by expressing both sides with a common base, revealing x = -4.

Question

(4/7)^x=2401/256

The user has presented an exponential equation:

The goal is to find the value of

Step 1: Analyze the base and the result.
The base of the exponent is

. We need to determine if the fraction

can be expressed as a power of

Step 2: Express the numerator (2401) and the denominator (256) as powers of their respective prime factors, or powers of 4 and 7.

Let’s check the powers of 4:

So, the denominator is

Let’s check the powers of 7:

So, the numerator is

Step 3: Rewrite the right-hand side of the equation using these powers.

Using the exponent rule

Step 4: Substitute this back into the original equation.

Step 5: Relate the bases.
Notice that the base on the left side,

, is the reciprocal of the base on the right side,

.
We know that

. Therefore,

Substitute this into the left side of the equation:

Using the exponent rule

Step 6: Solve for

.
Since the bases are equal and greater than 1, the exponents must be equal:

Step 7: Verification (Optional).
If

This matches the right side of the original equation.

The final answer is