Question

If \(\log_{10} 11 = a\) then \(\log_{10} \left(\frac{1}{110}\right)\) is equal to?

• A. \(-a\)
• B. \((1 + a)^{-1}\)
• C. \(\frac{1}{10a}\)
• D. \(-(a + 1)\)

Answer 1

The Correct Option is D

Given: \(\log_{10} 11 = a\)
We need to find \(\log_{10} \left(\frac{1}{110}\right)\).
We can express \(\frac{1}{110}\) as:
\[\frac{1}{110} = \frac{1}{11 \times 10} = \frac{1}{11} \times \frac{1}{10}\]
Using logarithm properties:
\[\log_{10} \left(\frac{1}{110}\right) = \log_{10} \left(\frac{1}{11 \times 10}\right) = \log_{10} \left(\frac{1}{11}\right) + \log_{10} \left(\frac{1{10}\right)
\]
We know:
\[\log_{10} \left(\frac{1}{x}\right) = -\log_{10} x\]
So:
\[\log_{10} \left(\frac{1}{11}\right) = -\log_{10} 11 = -a\]
And:
\[\log_{10} \left(\frac{1}{10}\right) = -\log_{10} 10 = -1\]
Adding these together:
\[\log_{10} \left(\frac{1}{110}\right) = -a + (-1) = -(a + 1)\]
Thus, the answer is:
D) \(-(a + 1)\)