Question

If \( 2^x = 5^{y} = 10^{-z} \), then the value of \( \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \) is:

• A. 3
• B. 5
• C. 0
• D. -2

Hint:

For equations involving exponents, converting to logarithmic form often makes the relationship between variables more manageable.

Answer 1

The Correct Option is C

We are given that: \[ 2^x = 5^y = 10^{-z} \] From the equation \( 2^x = 10^{-z} \), taking the logarithm base 10 of both sides: \[ x \log 2 = -z \log 10 \quad \Rightarrow \quad x \log 2 = -z \] Similarly, from \( 5^y = 10^{-z} \), we get: \[ y \log 5 = -z \log 10 \quad \Rightarrow \quad y \log 5 = -z \] Now we can use these relationships to express \( x, y, z \) and calculate \( \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \). After calculation, the final result is \( 0 \).