If \log_3 x - 4\log_x 3 = 3 for x > 1, what is the value of x? — JAMB UTME

If $\log_3 x - 4\log_x 3 = 3$ for $x > 1$ , what is the value of $x$ ?

Answer: 81

The value of

x

is 81.

Applying the change of base identity

\log_x 3 = \frac{1}{\log_3 x}

transforms the equation into

\log_3 x - \frac{4}{\log_3 x} = 3

. Substituting

u = \log_3 x

yields

u^2 - 3u - 4 = 0

, which factors as

(u - 4)(u + 1) = 0

. Given

x > 1

u

must be positive, giving

u = 4

. Converting back to exponential form gives

x = 3^4 = 81

Apply the change of base identity

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

to the term

\log_x 3

The term becomes

\frac{1}{\log_3 x}

, so the equation is

\log_3 x - \frac{4}{\log_3 x} = 3

To express all logarithmic terms in terms of a single common base.

Introduce a substitution variable

u = \log_3 x

The equation reduces to

u - \frac{4}{u} = 3

Simplifies the equation to a manageable algebraic structure.

Multiply the entire equation by

u

and rearrange terms.

u^2 - 3u - 4 = 0

Converts the rational expression into standard quadratic form.

Factor the quadratic equation.

(u - 4)(u + 1) = 0 \implies u = 4 \text{ or } u = -1

Finds the candidate values for

u

Filter out invalid roots based on the domain restriction

x > 1

and solve for

x

Since

x > 1

\log_3 x > 0

, so

u = 4

. Thus

\log_3 x = 4 \implies x = 3^4 = 81

Excludes extraneous solutions and evaluates the final exponentiation.

Logarithmic Change of Base and Quadratic Reduction

Question