# DBT JRF 2022 — Question 25

Exam: DBT JRF 2022 · Section: A · Topic: Aptitude › Algebra & Logarithms · Difficulty: Medium

## Question

If $2^x = 3^y = 6^z = K$ , then the relationship between x, y, and z is given by:

## Options

- **A.** x+y = z
- **B.** xy = z
- **C.** $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$
- **D.** $x^z = y^x = z$

## Answer

Option C

## Explanation

Algebra with exponents: If $2^x = 3^y = 6^z = K$ , find the relationship between x, y, z: Express each base in terms of K using logarithms: $2 = K^{1/x}$ , $3 = K^{1/y}$ , $6 = K^{1/z}$ Since $6 = 2 \times 3$ : $K^{1/z} = K^{1/x} \times K^{1/y} = K^{\frac{1}{x} + \frac{1}{y}}$ Equating exponents: $\frac{1}{z} = \frac{1}{x} + \frac{1}{y}$ Verification: $\log 6 = \log 2 + \log 3$ , so $\frac{\log K}{z} = \frac{\log K}{x} + \frac{\log K}{y}$ ✔ Answer: $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$ (Option 3).

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