The question is from Log and Exponents. A question that combines logarithm with Algebra. We need to find out a logarithm in terms of a and b. With every extra hour you log in for this topic, it becomes exponentially simpler. CAT Logarithms and Exponents is a favorite in CAT Exam, and appears more often than expected in the CAT Quantitative Aptitude section in the CAT Exam

Question 4: If log_{12}27 = a, log_{9}16 = b, find log_{8}108.

- \\frac{2(a + 3)}{3b}\\)
- \\frac{2(a + 3)}{3a}\\)
- \\frac{2(b + 3)}{3a}\\)
- \\frac{2(b + 3)}{3b}\\)

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log_{8}108 = log_{8}(4 * 27)

log_{8}108 = log_{8}4 + log_{8}27

=> log_{8}4 = \\frac{2}{3}\\)

log_{8}27 = \\frac{log_16{27}}{log_16{8}}\\)

log_{8}27 = \\frac{log_16{27}}{3/4}\\)

log_{8}27 = \\frac{4}{3}\\) log_{16}27

log_{8}27 = \\frac{4}{3}\\) \\frac{log_9{27}}{log_9{16}}\\)

log_{8}27 = \\frac{4}{3}\\) \\frac{\\frac{3}{2}}{log_9{16}}\\)

log_{8}27 = 2 * log_{16}9

log_{9}16 = b

log_{16}9 = \\frac{1}{b}\\)

log_{8}27 = \\frac{2}{b}\\)

log_{8}108 = \\frac{2}{3}\\) + \\frac{2}{b}\\)

= 2(\\frac{1}{3}\\) + \\frac{1}{b}\\))

= \\frac{2(b + 3)}{3b}\\).

The question is **"find log _{8}108."**

Choice D is the correct answer.

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