If x and y are real numbers greater than 1 satisfying the system of equations \log_x y + \… — JAMB UTME

If $x$ and $y$ are real numbers greater than $1$ satisfying the system of equations $\log_x y + \log_y x = \frac{5}{2}$ and $xy = 64$ with $x > y$ , find the value of $x - y$ .

Answer: 12

Answer

The value of

x - y

is 12.

Using the change of base identity

\log_y x = \frac{1}{\log_x y}

, the equation

\log_x y + \log_y x = \frac{5}{2}

converts to

u + \frac{1}{u} = \frac{5}{2}

for

u = \log_x y

. Solving the quadratic equation

2u^2 - 5u + 2 = 0

yields

u = \frac{1}{2}

u = 2

. Because

x > y > 1

, we must have

\log_x y < 1

, selecting

u = \frac{1}{2} \implies x = y^2

. Substituting into

xy = 64

gives

y^3 = 64 \implies y = 4

and

x = 16

. Therefore,

x - y = 16 - 4 = 12

Step-by-Step Solution

Apply the reciprocal change of base identity

Rewrite

\log_y x

\frac{1}{\log_x y}

, yielding

\log_x y + \frac{1}{\log_x y} = \frac{5}{2}

According to the change of base formula,

\log_y x = \frac{\log_x x}{\log_x y} = \frac{1}{\log_x y}

Solve the quadratic equation in terms of

u = \log_x y

Substituting

u = \log_x y

gives

u + \frac{1}{u} = \frac{5}{2} \implies 2u^2 - 5u + 2 = 0

, which factors into

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

, yielding

u = \frac{1}{2}

u = 2

Multiplying through by

2u

clears fractions and forms a standard quadratic equation.

Select the valid root using given inequality constraints

Since

x > y > 1

, taking the logarithm base

x

yields

\log_x x > \log_x y \implies 1 > \log_x y

. Thus

u = \frac{1}{2}

, which means

y = x^{1/2}

x = y^2

The condition

x > y

restricts the logarithm of

y

base

x

to be strictly less than

1

Substitute into the product equation to find

x

and

y

Substituting

x = y^2

into

xy = 64

gives

y^3 = 64 \implies y = 4

. Consequently,

x = 4^2 = 16

Combining the relation

x = y^2

with

xy = 64

enables single-variable cubic solution.

Calculate the required difference

x - y

16 - 4 = 12

Direct subtraction of the derived values

x = 16

and

y = 4

Key Concept

Logarithmic Change of Base Reciprocal Property

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