The question is from Log and Exponents. This questions is about Logarithm - Different bases. We need to find out the value of log in terms of a and b based on given another two log values. 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 6: log_{5}x = a (This should be read as log X to the base 5 equals a) log_{20}x = b. What is log_{x}10?

- \\frac{a + b}{2ab}\\)
- (a + b) * 2ab
- \\frac{2ab}{a + b}\\)
- \\frac{a + b}{2}\\)

\\frac{a + b}{2ab}\\)

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Given, log_{5}x = a

log_{20}x = b

log_{x}5 = \\frac{1}{a}\\)

log_{x}20 = \\frac{1}{b}\\)

log_{x}\\frac{100}{5}\\) = \\frac{1}{b}\\)

log_{x}100 - log_{n}5 = \\frac{1}{b}\\)

=> 2log_{x}10 - \\frac{1}{a}\\) = \\frac{1}{b}\\)

log_{x}10 = \\frac{1}{2}\\)(\\frac{1}{a}\\) + \\frac{1}{b}\\))

log_{x}10 = \\frac{a + b}{2ab}\\)

The question is **"What is log _{x}10?"**

Choice A is the correct answer.

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