Logarithmic Range for x

CAT 2017 Slot 2 · Quantitative Ability · Hard · Exponents & Logarithms

This is a hard Quantitative Ability question from the CAT 2017 Slot 2 paper. It tests Exponents & Logarithms. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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If $x$ is a real number such that $\log_3 5 = \log_5(2 + x)$ , then which of the following is true?

A.
$0 < x < 3$
B.
$23 < x < 30$
C.
$x > 30$
D.
$3 < x < 23$

Answer

Explanation

We know $1 < \log_3 5 < 2$ , because $3^1 = 3 < 5 < 9 = 3^2$ .

Let $k = \log_3 5$ , so $1 < k < 2$ .

Given $\log_5(2 + x) = k \implies 2 + x = 5^k$ .

Since $1 < k < 2$ : $5^1 < 5^k < 5^2 \implies 5 < 2 + x < 25$ $3 < x < 23$

Thus, $3 < x < 23$ .

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