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Quantitative Ability and Data Interpretation Question on Logarithms

If log⁡1011=a\log_{10} 11 = a then log⁡10(1110)\log_{10} \left(\frac{1}{110}\right) is equal to?

A

−a-a

B

(1+a)−1(1 + a)^{-1}

C

110a\frac{1}{10a}

D

−(a+1)-(a + 1)

Answer

−(a+1)-(a + 1)

Explanation

Solution

Given: log⁡1011=a\log_{10} 11 = a
We need to find log⁡10(1110)\log_{10} \left(\frac{1}{110}\right).
We can express 1110\frac{1}{110} as:
1110=111×10=111×110\frac{1}{110} = \frac{1}{11 \times 10} = \frac{1}{11} \times \frac{1}{10}
Using logarithm properties:

We know:
log⁡10(1x)=−log⁡10x\log_{10} \left(\frac{1}{x}\right) = -\log_{10} x
So:
log⁡10(111)=−log⁡1011=−a\log_{10} \left(\frac{1}{11}\right) = -\log_{10} 11 = -a
And:
log⁡10(110)=−log⁡1010=−1\log_{10} \left(\frac{1}{10}\right) = -\log_{10} 10 = -1
Adding these together:
log⁡10(1110)=−a+(−1)=−(a+1)\log_{10} \left(\frac{1}{110}\right) = -a + (-1) = -(a + 1)
Thus, the answer is:
D) −(a+1)-(a + 1)