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CAT 2018 Slot 1 — QA Question 22

LogarithmsEasy

If x is a positive quantity such that 2x = 3log5(2) , then x is equal to

Answer & solution

Correct answer: 1 + log 5 3 5

  • A

    1 + log3(53)

  • B

    log59

  • C

    log58

  • 1 + log5(35)

Solution

Easy

Use the identity alog⁡bc=clog⁡baa^{\log_b c} = c^{\log_b a} to swap the base 33 for base 22, read off xx as a clean log, then rewrite that log to match an option.

1

Swap the base. Apply 3log⁡52=2log⁡533^{\log_5 2} = 2^{\log_5 3}:

2x=3log⁡52=2log⁡53(since alog⁡bc=clog⁡ba)⇒ x=log⁡53\begin{aligned} &2^{x} = 3^{\log_5 2} = 2^{\log_5 3} \quad\text{(since }a^{\log_b c}=c^{\log_b a})\\ &\Rightarrow\ x = \log_5 3 \end{aligned}
2

Match to an option. Rewrite option (d):

1+log⁡5 ⁣(35)=1+log⁡53−log⁡55⇒ =1+log⁡53−1=log⁡53\begin{aligned} &1 + \log_5\!\left(\tfrac35\right) = 1 + \log_5 3 - \log_5 5\\ &\Rightarrow\ = 1 + \log_5 3 - 1 = \log_5 3 \end{aligned}

This equals xx, while option (a) gives log⁡35\log_3 5, (b) gives 2log⁡532\log_5 3, and (c) gives 3log⁡523\log_5 2 — none of which equal log⁡53\log_5 3.

x=log⁡53=1+log⁡5 ⁣(35)x = \log_5 3 = 1 + \log_5\!\left(\dfrac35\right)

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CAT 2018 Slot 1 QA Q22: If x is a positive quantity such that 2 x = 3 log 5 2 , then x is equal to — Solution | TheCATExam