If \log_5 x + 2\log_x 5 = 3, what is the sum of all possible real values of x? — JAMB UTME

If $\log_5 x + 2\log_x 5 = 3$ , what is the sum of all possible real values of $x$ ?

The sum of all possible real values of

x

is 30.

Applying the change of base identity

\log_x 5 = \frac{1}{\log_5 x}

converts the equation to

\log_5 x + \frac{2}{\log_5 x} = 3

. Setting

y = \log_5 x

yields

y^2 - 3y + 2 = 0

, giving solutions

y = 1

and

y = 2

. Converting back to

x

gives

x = 5^1 = 5

and

x = 5^2 = 25

. The sum of these values is

5 + 25 = 30

Apply the change of base identity

\log_x 5 = \frac{1}{\log_5 x}

The equation becomes

\log_5 x + \frac{2}{\log_5 x} = 3

Converting all logarithmic terms to base 5 allows substitution and simplification.

Substitute

y = \log_5 x

and form a quadratic equation.

y + \frac{2}{y} = 3 \implies y^2 - 3y + 2 = 0

Multiplying through by

y

transforms the logarithmic equation into a standard quadratic form.

Solve the quadratic equation for

y

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

Factoring determines the values of

y = \log_5 x

Convert back to

x

using exponentiation and calculate the sum.

For

y = 1

x = 5^1 = 5

. For

y = 2

x = 5^2 = 25

. Sum

= 5 + 25 = 30

Definition of logarithm

\log_a b = c \iff a^c = b

gives the values of

x

Logarithm Change of Base Formula and Quadratic Substitution

Estimated Time:1m 30s

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