Simplifying Logarithmic Expression

CAT 2019 Slot 1 · QA · Easy · Logarithm

If log1281=p\log_{12} 81 = p, then 3(4p4+p)3\left(\frac{4-p}{4+p}\right) is equal to:

  1. A.

    log28\log_2 8

  2. B.

    log68\log_6 8

  3. C.

    log416\log_4 16

  4. D.

    log616\log_6 16

Answer

B

Explanation

p=log1281=4log3log12=4log32log2+log3p = \log_{12} 81 = \frac{4\log 3}{\log 12} = \frac{4\log 3}{2\log 2 + \log 3}. 4p=44log32log2+log3=8log22log2+log34 - p = 4 - \frac{4\log 3}{2\log 2 + \log 3} = \frac{8\log 2}{2\log 2 + \log 3}. 4+p=4+4log32log2+log3=8log2+8log32log2+log34 + p = 4 + \frac{4\log 3}{2\log 2 + \log 3} = \frac{8\log 2 + 8\log 3}{2\log 2 + \log 3}. 4p4+p=8log28(log2+log3)=log2log6=log62\frac{4-p}{4+p} = \frac{8\log 2}{8(\log 2 + \log 3)} = \frac{\log 2}{\log 6} = \log_6 2. 3(4p4+p)=3log62=log683 \left(\frac{4-p}{4+p}\right) = 3 \log_6 2 = \log_6 8.

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