CAT 2019 Slot 1 · QA · Easy · Logarithm
If log1281=p\log_{12} 81 = plog1281=p, then 3(4−p4+p)3\left(\frac{4-p}{4+p}\right)3(4+p4−p) is equal to:
log28\log_2 8log28
log68\log_6 8log68
log416\log_4 16log416
log616\log_6 16log616
B
p=log1281=4log3log12=4log32log2+log3p = \log_{12} 81 = \frac{4\log 3}{\log 12} = \frac{4\log 3}{2\log 2 + \log 3}p=log1281=log124log3=2log2+log34log3. 4−p=4−4log32log2+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=4−2log2+log34log3=2log2+log38log2. 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}4+p=4+2log2+log34log3=2log2+log38log2+8log3. 4−p4+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 24+p4−p=8(log2+log3)8log2=log6log2=log62. 3(4−p4+p)=3log62=log683 \left(\frac{4-p}{4+p}\right) = 3 \log_6 2 = \log_6 83(4+p4−p)=3log62=log68.
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