CAT Quantitative Aptitude Questions | CAT Logarithms and Exponents

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: log₅x = a (This should be read as log X to the base 5 equals a) log₂₀x = b. What is log_x10?

1. $\frac{a + b}{2ab}\\$
2. (a + b) * 2ab
3. $\frac{2ab}{a + b}\\$
4. $\frac{a + b}{2}\\$

Explanatory Answer

Method of solving this CAT Question from Logs and Exponents: When we change the base, Log questions become far tougher.

Given, log₅x = a
log₂₀x = b
log_x5 = $\frac{1}{a}\\$
log_x20 = $\frac{1}{b}\\$
log_x$\frac{100}{5}\\$ = $\frac{1}{b}\\$
log_x100 - log_n5 = $\frac{1}{b}\\$
=> 2log_x10 - $\frac{1}{a}\\$ = $\frac{1}{b}\\$
log_x10 = $\frac{1}{2}\\$($\frac{1}{a}\\$ + $\frac{1}{b}\\$)
log_x10 = $\frac{a + b}{2ab}\\$

The question is "What is log_x10?"

Hence, the answer is "$\frac{a + b}{2ab}\\$".

Choice A is the correct answer.