CAT 2020 Question Paper | Quants Slot 3

Logarithms is one of the most commonly tested topics in CAT exam. This topic plays a pivotal role in your CAT preparation. Questions from Logarithms have appeared consistently for several years in CAT exam. This question tests your understanding of the properties of Logarithms. If you're thorough with your fundamentals, then you're through. CAT Exam is known to test your fundamentals rather than high funda ideas. CAT Logarithms and Exponents is a favorite in CAT Exam, with every extra hour you log in for this topic, it becomes exponentially simpler.

Question 3 : Let log_a30 = A, log_a$\frac{5}{3}$ = -B and log₂a = $\frac{1}{3}$, then log₃a equals

Video Explanation

Explanatory Answer

Let log_a30 = A be equation 1.
and log_a$\frac{5}{3}$ = -B be equation 2.

We are told that log₂a = $\frac{1}{3}$
log₂a = $\frac{1}{3}$ implies log_a2 = 3

Dividing eqn(1) and eq(2) by log_a2 = 3
We get; log₂30 = $\frac{A}{3}$ and
log₂$\frac{5}{3}$ = -$\frac{B}{3}$

log₂30 = log₂(5 × 3 × 2)
$\frac{A}{3}$ = log₂5 + log₂3 + log₂2
$\frac{A}{3}$ = log₂5 + log₂3 + 1, let's call this equation 3

log₂$\frac{5}{3}$ = log₂5 - log₂3
-$\frac{B}{3}$ = log₂5 - log₂3, let's call this equation 4

Solving equations 3 and 4 we get, log₂3 = $\frac{A+B-3}{6}$
Dividing this by log₂a = $\frac{1}{3}$ we get,
log₂a = $\frac{A+B-3}{2}$

log₂a = $\frac{A+B-3}{2}$ implies log_a2 = $\frac{2}{A+B-3}$

The question is "Let log_a30 = A, log_a$\frac{5}{3}$ = -B and log₂a = $\frac{1}{3}$, then log₃a equals"

Hence, the answer is, "$\frac{2}{A+B-3}$"

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