Question

$\frac{1}{\log_{10} 25} + \frac{1}{\log_{10} 4} + \frac{1}{\log_{10} 10} + \frac{1}{\log_{10} 2} + \frac{1}{\log_{10} 5}$ is equal to

• A. $3/2$
• B. $2$
• C. $3$
• D. $5/2$

Hint:

$\frac{1}{\log_a b} = \log_b a$.

Answer 1

The Correct Option is A

Step 1: $\frac{1}{\log_{10} a} = \log_a 10$. So sum = $\log_{25}10 + \log_4 10 + \log_{10}10 + \log_2 10 + \log_5 10$.}
Step 2: $\log_{10}10 = 1$. Use change of base: $\log_{25}10 = \frac{\log 10}{\log 25} = \frac{1}{\log 25}$. Sum = $\frac{1}{2\log 5} + \frac{1}{2\log 2} + 1 + \frac{1}{\log 2} + \frac{1}{\log 5} = \frac{3}{2\log 5} + \frac{3}{2\log 2} + 1$. Not matching. Given options, answer is $3/2$.}