If 27^{x - 1} = 9^{x + 1}, determine the value of x. — JAMB UTME

If $27^{x - 1} = 9^{x + 1}$ , determine the value of $x$ .

Answer: 5

The value of

x

is 5.

Rewriting 27 as

3^3

and 9 as

3^2

transforms the given equation into

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

. Equating exponents gives

3x - 3 = 2x + 2

, which simplifies directly to

x = 5

Express numbers in terms of a common base

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

Both 27 and 9 are powers of 3, allowing reduction to a single base.

Apply power of a power index law

3^{3x - 3} = 3^{2x + 2}

Multiply the base power by the expression in the exponent:

3 \times (x - 1) = 3x - 3

and

2 \times (x + 1) = 2x + 2

Equate exponents and solve for

x

3x - 3 = 2x + 2 \implies x = 5

Equal bases imply that the index powers must be equal.

Equating exponential expressions using a common base

Question