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

Given: \[ 2^x = 5^y = 10^{-z} \] From \( 2^x = 10^{-z} \), taking log base 10: \[ x \log 2 = -z \log 10 \quad \Rightarrow \quad x \log 2 = -z \] From \( 5^y = 10^{-z} \): \[ y \log 5 = -z \log 10 \quad \Rightarrow \quad y \log 5 = -z \] Calculate \( x, y, z \) to find \( \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \). The final result is \( 0 \).