If \log_3 x - 2 \log_x 3 = 1, what is the sum of all valid solutions for x? — JAMB UTME

If $\log_3 x - 2 \log_x 3 = 1$ , what is the sum of all valid solutions for $x$ ?

$\frac{28}{3}$ Cevap
B
$12$
C
$1$
D
$9$

Cevap

The sum of all valid solutions for

x

\frac{28}{3}

By using the change of base identity

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

and substituting

u = \log_3 x

, the given equation simplifies to

u - \frac{2}{u} = 1

. Multiplying by

u

gives the quadratic equation

u^2 - u - 2 = 0

, which factors as

(u - 2)(u + 1) = 0

. This yields

u = 2

u = -1

. Converting back to

x

gives

x = 3^2 = 9

and

x = 3^{-1} = \frac{1}{3}

. Both values satisfy domain constraints for logarithmic base (

x > 0

and

x \neq 1

). Adding these values gives

9 + \frac{1}{3} = \frac{28}{3}

Adım Adım Çözüm

Apply the change of base formula to express the equation with a common logarithmic base

Using

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

, the equation becomes

\log_3 x - \frac{2}{\log_3 x} = 1

Change of base allows all terms containing

x

to be represented in terms of a single logarithmic expression.

Substitute

u = \log_3 x

and clear the fraction to form a quadratic equation

u - \frac{2}{u} = 1 \implies u^2 - u - 2 = 0

Multiplying through by

u

(where

u \neq 0

) transforms the logarithmic relationship into a standard quadratic algebraic equation.

Factor the quadratic equation to find the values of

u

(u - 2)(u + 1) = 0 \implies u = 2 \text{ or } u = -1

Factoring isolates the linear roots for the substituted variable

u

Solve for

x

from

u = \log_3 x

and verify validity within the domain

For

u = 2

\log_3 x = 2 \implies x = 3^2 = 9

.
For

u = -1

\log_3 x = -1 \implies x = 3^{-1} = \frac{1}{3}

.
Both solutions are positive and

x \neq 1

, so both are valid.

Converting from logarithmic form to exponential form retrieves the original variable

x

Calculate the sum of all valid solutions

Sum =

9 + \frac{1}{3} = \frac{27}{3} + \frac{1}{3} = \frac{28}{3}

The question requires finding the total sum of all permissible real solutions for

x

Anahtar Kavram

Logarithmic Change of Base and Equations Reducible to Quadratics

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