Will log accept me?

Algebra Level 3

It is true that 110 > 3 ⟹ log ⁡ 10 > log ⁡ 3 ⟹ 0 > log ⁡ 3 − 1 \begin{array}{c}110 > 3 \implies \log 10 > \log 3 \implies 0 > \log 3 -1 \end{array} Then it must be true that 44 > 3 ⟹ log ⁡ 4 − 1 > log ⁡ 3 − 1 ⟹ log ⁡ 4 − 1 log ⁡ 3 − 1 > 1 . \begin{array}{c}44 > 3 \implies \log 4 - 1 > \log 3 -1 \implies \dfrac{\log 4 -1}{\log 3-1} > 1 \,.\end{array} If possible try to solve without using calculators .

No Yes

Naren Bhandari by Naren Bhandari , 21, India

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1 solution

Brian Moehring
Aug 3, 2018

We can just use the first derivation to see that log ⁡ 3 − 1 < 0 \log 3 - 1 < 0 is negative, and if we divide both sides of log ⁡ 4 − 1 > log ⁡ 3 − 1 \log 4 - 1 > \log 3 - 1 by a negative number, we must change the direction of the inequality, so it gives log ⁡ 4 − 1 log ⁡ 3 − 1 < log ⁡ 3 − 1 log ⁡ 3 − 1 = 1. \frac{\log 4 - 1}{\log 3 - 1} < \frac{\log 3 - 1}{\log 3 - 1} = 1.

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