Find the real value of x that satisfies the exponential equation 5^{2x - 1} \times 25^{x +… — JAMB UTME

Find the real value of

x

that satisfies the exponential equation

5^{2x - 1} \times 25^{x + 1} = 125^{x + 2}

Answer: 5

The value of

x

is 5.

By converting all terms to base 5 (

25 = 5^2

and

125 = 5^3

), the equation becomes

5^{2x-1} \times 5^{2(x+1)} = 5^{3(x+2)}

. Simplifying exponents gives

5^{2x-1} \times 5^{2x+2} = 5^{3x+6}

. Adding the left-hand powers results in

5^{4x+1} = 5^{3x+6}

. Equating the exponents yields

4x + 1 = 3x + 6

, leading directly to

x = 5

Express all terms with a common base of 5

5^{2x-1} \times 5^{2x+2} = 5^{3x+6}

Since

25 = 5^2

and

125 = 5^3

, using index laws

(a^m)^n = a^{mn}

allows all expressions to share base 5.

Apply the product rule of indices on the left side

5^{4x+1} = 5^{3x+6}

According to the product law

a^m \times a^n = a^{m+n}

, the powers are added:

(2x-1) + (2x+2) = 4x+1

Equate powers of equal bases to solve for x

x = 5

Since bases are equal, exponents must be equal:

4x + 1 = 3x + 6 \implies 4x - 3x = 6 - 1 \implies x = 5

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