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CAT 2017 Slot 2 — QA Question 22

LogarithmsEasy

If x is a real number such that log35 = log5(2 + x), then which of the following is true?

Answer & solution

Correct answer: 3 < x < 23

  • A

    0 < x < 3

  • B

    23 < x < 30

  • C

    x > 30

  • 3 < x < 23

Solution

Easy

Bound log⁡35\log_3 5 between consecutive integers, then transfer that bound onto log⁡5(2+x)\log_5(2+x) since the two are equal. Converting the logarithmic inequality to exponential form gives the range of xx.

1

Bound the common value. Since 3<5<93 < 5 < 9, taking log⁡3\log_3 traps log⁡35\log_3 5 between 11 and 22.

log⁡33<log⁡35<log⁡39⇒ 1<log⁡35<2\begin{aligned} &\log_3 3 < \log_3 5 < \log_3 9\\ &\Rightarrow\ 1 < \log_3 5 < 2 \end{aligned}
2

Transfer to the other log. The equation makes log⁡5(2+x)\log_5(2+x) equal to log⁡35\log_3 5, so it inherits the same bounds; rewrite endpoints as base-55 logs.

1<log⁡5(2+x)<2(from step 1)⇒ log⁡55<log⁡5(2+x)<log⁡525\begin{aligned} &1 < \log_5(2+x) < 2 \quad\text{(from step 1)}\\ &\Rightarrow\ \log_5 5 < \log_5(2+x) < \log_5 25 \end{aligned}
3

Exponentiate. Base 5>15>1 keeps the inequality direction; drop the logs and isolate xx.

5<2+x<25⇒ 3<x<23\begin{aligned} &5 < 2 + x < 25\\ &\Rightarrow\ 3 < x < 23 \end{aligned}
3<x<233 < x < 23

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CAT 2017 Slot 2 QA Q22: If x is a real number such that log 3 5 = log 5 (2 + x ), then which of the following is true? — Solution | TheCATExam