Question

\(\frac{1}{log_2 100} - \frac{1}{log_4 100} + \frac{1}{log_5 100} - \frac{1}{log_{10} 100} + \frac{1}{log_{20} 100} - \frac{1}{log_{25}100} + \frac{1}{log_{50} 100}- ?\)

Answer 1

Correct Answer - Option 4 : 1/2

Concept Used:

\(\frac{1}{log_b a}\)= \(log_a b\)

\(log_{a^p} b = \frac{1}{p} log_a b\)

Calculation:

\(\frac{1}{log_2 100} - \frac{1}{log_4 100} + \frac{1}{log_5 100} - \frac{1}{log_{10} 100} + \frac{1}{log_{20} 100} - \frac{1}{log_{25}100} + \frac{1}{log_{50} 100}\)

\(⇒{log_{100} 2} - {log_{100} 4} + {log_{100} 5} - {log_{100} 10} + {log_{100} 20} - {log_{100} 25} + {log_{100} 50}\)

⇒ \({log_{100} (2/4 × 5/10 × 20/25 × 50)} \)

⇒ \({log_{100} 10}\)

⇒ \({log_{10^2} 10}\)        

⇒ 1/2                [\(log_{10}10 = 1\)]