Question

1/log2100-1/log4100+1/log5100-1/log10100+1/log20100-1/log25100+1/log50100=?

• A. \(\frac 12\)
• B. 0
• C. 10
• D. -4

Answer 1

The Correct Option is A

To solve the given expression, we start by simplifying each term using the change of base formula for logarithms: \(\frac{1}{\log_b a} = \log_a b\).

So, the expression becomes: 

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

Combine the terms using the properties of logarithms:

\(\log_{100} \left(\frac{2 \cdot 5 \cdot 20 \cdot 50}{4 \cdot 10 \cdot 25}\right)\)

Simplify the argument of the logarithm:

\(\frac{2 \times 5 \times 20 \times 50}{4 \times 10 \times 25} = \frac{2 \times 5 \times 4 \times 5 \times 10 \times 5}{2^2 \times 5^2 \times 2 \times 5} = 2 \times 5 = 10\)

Thus, the expression simplifies to:

\(\log_{100} 10\)

Using the properties of logarithms, \(\log_{100} 10\) can be written as:

\(\frac{\log 10}{\log 100} = \frac{1}{2}\)

So, the result of the original expression is \(\frac{1}{2}\).

The correct answer is \(\frac{1}{2}\).