Q.4 For positive non–zero real variables 𝑥 and 𝑦, if
ln (𝑥 + 𝑦/2 ) = 1/2 [ln (𝑥) + ln (𝑦)]
hen, the value of    x/y+y/x is
(A) 1
(B) 1/2
(C) 2
(D) 4

The equation ln((x + y)/2) = 1/2 [ln(x) + ln(y)] for positive non-zero real variables x and y equates the logarithmic mean and the geometric mean, which forces x = y and leads to the result y/x + x/y = 2. This CSIR NET and GATE-style aptitude problem checks your understanding of logarithm properties and classical inequalities.

Problem statement

For positive non-zero real variables x and y, suppose

ln((x + y)/2) = 1/2 [ln(x) + ln(y)].

You are asked to find the value of:

y/x + x/y.

Step-by-step solution

1. Use logarithm properties

Start from the given equation:

ln((x + y)/2) = 1/2 [ln(x) + ln(y)].

Apply the property ln(a) + ln(b) = ln(ab). Then

1/2 [ln(x) + ln(y)] = 1/2 ln(xy).

So the equation becomes:

ln((x + y)/2) = 1/2 ln(xy).

2. Remove the logarithms

Multiply both sides by 2:

2 · ln((x + y)/2) = ln(xy).

Use the power rule c · ln(A) = ln(Ac) to write:

ln(((x + y)/2)2) = ln(xy).

Because the natural logarithm function is one-to-one on positive numbers, the arguments must match:

((x + y)/2)2 = xy.

3. Simplify the algebraic equation

Expand the left-hand side:

(x + y)2 / 4 = xy.

Multiply both sides by 4:

(x + y)2 = 4xy.

Expand the square:

x2 + 2xy + y2 = 4xy.

Rearrange the terms:

x2 + 2xy + y2 - 4xy = 0

x2 - 2xy + y2 = 0.

Recognize a perfect square:

(x - y)2 = 0.

Hence

x = y (since x and y are real).

4. Compute y/x + x/y

From x = y, we get:

y/x = 1 and x/y = 1.

Therefore,

y/x + x/y = 1 + 1 = 2.

So the required value is 2.

Checking with AM-GM inequality

The equation ((x + y)/2)2 = xy can be written as (x + y)/2 = √(xy), which states that the arithmetic mean of x and y equals their geometric mean. By the AM–GM inequality, this equality holds only when x = y.

Once we know x = y, the expression y/x + x/y clearly equals 2. This also shows that any value less than 2 for y/x + x/y is impossible when x, y > 0.

Option-wise explanation

Typical multiple-choice options for this question are:

• (A) 1
• (B) 1/2
• (C) 2
• (D) 4

Option (A) 1

This option is incorrect. For positive x and y, the expression y/x + x/y is always greater than or equal to 2 by AM–GM, with equality only when x = y. To get a value of 1, you would need either x = 0 or y = 0, which contradicts the condition that both variables are positive and non-zero.

Option (B) 1/2

This option is also incorrect. The minimum possible value of y/x + x/y under the constraint x, y > 0 is 2, so 1/2 is far too small and cannot occur.

Option (C) 2

This is the correct option. From the algebraic derivation, the condition on the logarithms forces x = y, and then y/x + x/y = 1 + 1 = 2. This exactly matches the minimum value predicted by AM–GM, confirming that 2 is the only valid answer.

Option (D) 4

This option is incorrect for the given equation. A value like 4 for y/x + x/y may arise in other variants of similar problems, for example where the logarithmic equation is changed (such as involving (x − y) instead of (x + y)), but not for the present condition ln((x + y)/2) = 1/2 [ln(x) + ln(y)].

Key logarithm properties used

• Product rule: ln(ab) = ln(a) + ln(b).
• Power rule: c · ln(A) = ln(Ac).
• One-to-one property: If ln(A) = ln(B) and A, B > 0, then A = B.

These basic properties convert the logarithmic equation into a simple algebraic equation that can be solved using expansion and factorization.

Proof relevance for competitive exams

Problems of the form ln((x + y)/2) = 1/2 (ln x + ln y) are common in GATE, CSIR NET, and other competitive exams to test conceptual clarity of logarithms and inequalities. They connect logarithmic identities with the AM–GM inequality in a compact way.

Mastering this pattern helps in quickly recognizing that equal arithmetic and geometric means imply equal variables, allowing you to evaluate expressions like y/x + x/y in just a couple of steps during the exam.

SEO keywords

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  Solve ln((x + y)/2) = 1/2 (ln x + ln y): Find y/x + x/y for Positive Real Variables

  For positive non-zero real variables x and y, consider the equation:

  Given: ln((x + y)/2) = 1/2 [ln(x) + ln(y)].

  We are asked to find the value of y/x + x/y.

  Step-by-step solution

  Using the logarithm property ln(a) + ln(b) = ln(ab), the right-hand side becomes:

  1/2 [ln(x) + ln(y)] = 1/2 ln(xy).

  So the equation is:

  ln((x + y)/2) = 1/2 ln(xy).

  Rewrite the right side as a single logarithm:

  1/2 ln(xy) = ln((xy)^1/2) = ln(√(xy)).

  Thus:

  ln((x + y)/2) = ln(√(xy)).

  Because the natural logarithm is one-to-one for positive arguments, we can equate the insides:

  (x + y)/2 = √(xy).

  Multiply both sides by 2:

  x + y = 2√(xy).

  Expressing y in terms of x

  Let k = y/x (k > 0), so y = kx.

  Substitute y = kx into x + y = 2√(xy):

  x + kx = 2√(x · kx).

  Factor and simplify:

  x(1 + k) = 2x√k.

  Since x > 0, divide both sides by x:

  1 + k = 2√k.

  Square both sides:

  (1 + k)² = 4k.

  1 + 2k + k² = 4k.

  Rearrange:

  k² – 2k + 1 = 0.

  (k – 1)² = 0.

  So k = 1, which means y/x = 1 and hence y = x.

  Finding y/x + x/y

  If y = x, then:

  y/x = 1 and x/y = 1.

  Therefore:

  y/x + x/y = 1 + 1 = 2.

  So, the required value is:

  y/x + x/y = 2.

  Explaining each option

  The options are:

  • (A) 1
  • (B) 1/2
  • (C) 2
  • (D) 4

  Option (A) 1: This is incorrect. For positive x and y, the AM-GM inequality gives x/y + y/x ≥ 2, with equality only when x = y. A value of 1 is smaller than this minimum and would violate the inequality.

  Option (B) 1/2: This is also incorrect, because it is even less than 1 and definitely less than the minimum possible value 2 of x/y + y/x for positive x and y.

  Option (C) 2: This is correct. From the equation ln((x + y)/2) = 1/2 (ln x + ln y), we derived that x = y, which directly gives y/x + x/y = 2.

  Option (D) 4: This is incorrect for this specific equation. A value like 4 may appear in similar variants where the logarithmic equation is different (for example, involving x – y instead of x + y), but not in the given problem.

  Concept link: logarithmic mean and geometric mean

  The condition ln((x + y)/2) = 1/2 [ln(x) + ln(y)] implies:

  (x + y)/2 = √(xy),

  which means the arithmetic mean equals the geometric mean. For positive numbers, this happens only when x = y. This is a classic application of logarithm properties and the AM-GM inequality, frequently tested in competitive exams like GATE and CSIR NET.

  Secondary keywords: positive non-zero real variables, y/x + x/y value, logarithmic mean geometric mean