What is the positive integer solution to the logarithmic equation \log_2 x - 3\log_x 2 = 2… — JAMB UTME

What is the positive integer solution to the logarithmic equation $\log_2 x - 3\log_x 2 = 2$ ?

The positive integer solution is

8

Using the change of base identity

\log_x 2 = \frac{1}{\log_2 x}

, we rewrite the equation as

u - \frac{3}{u} = 2

where

u = \log_2 x

. Rearranging yields the quadratic

u^2 - 2u - 3 = 0

, which factors as

(u - 3)(u + 1) = 0

. This gives

u = 3

u = -1

. Converting back to

x = 2^u

, we get

x = 2^3 = 8

x = 2^{-1} = \frac{1}{2}

. Since the question asks for the positive integer solution, the correct answer is 8.

Apply the change of base rule to express

\log_x 2

in terms of base 2.

\log_2 x - \frac{3}{\log_2 x} = 2

The change of base identity states that

\log_b a = \frac{1}{\log_a b}

Substitute

u = \log_2 x

into the equation and clear the fraction.

u - \frac{3}{u} = 2 \implies u^2 - 2u - 3 = 0

Multiplying through by

u

(where

u \neq 0

) transforms the equation into standard quadratic form.

Factor the quadratic equation to solve for

u

(u - 3)(u + 1) = 0 \implies u = 3 \text{ or } u = -1

Finding the roots of the quadratic equation in terms of

u

Convert back to

x

using

x = 2^u

and select the positive integer solution.

For

u = 3

x = 2^3 = 8

. For

u = -1

x = 2^{-1} = \frac{1}{2}

The question asks specifically for the positive integer solution, which is 8.

Logarithmic Change of Base and Quadratic Reduction

Question