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CAT 2020 Slot 3 — QA Question 19

LogarithmsEasy

2×4×8×16(log24)2(log48)3(log816)4 equals

Answer & solution

Correct answer: 24

Answer: 24

Solution

Easy

Write every base and argument as a power of 22. The numerator becomes a single power of 22; each logarithm in the denominator simplifies via log⁡2a2b=ba\log_{2^a}2^b=\tfrac{b}{a}. Then collect powers.

1

Numerator as a power of 22.

2×4×8×16=21⋅22⋅23⋅24=210\begin{aligned} &2\times 4\times 8\times 16=2^{1}\cdot 2^{2}\cdot 2^{3}\cdot 2^{4}=2^{10} \end{aligned}
2

Simplify each logarithm. Using log⁡2a2b=ba\log_{2^a}2^b=\dfrac{b}{a}.

log⁡24=log⁡222=2log⁡48=log⁡2223=32log⁡816=log⁡2324=43\begin{aligned} &\log_2 4=\log_2 2^2=2\\ &\log_4 8=\log_{2^2}2^3=\tfrac{3}{2}\\ &\log_8 16=\log_{2^3}2^4=\tfrac{4}{3} \end{aligned}
3

Denominator. Raise each to its given power.

(log⁡24)2(log⁡48)3(log⁡816)4=22⋅(32)3⋅(43)4⇒ =22⋅3323⋅2834(44=28)⇒ =22+8⋅3323⋅34=21023⋅3\begin{aligned} &(\log_2 4)^2(\log_4 8)^3(\log_8 16)^4=2^{2}\cdot\left(\tfrac{3}{2}\right)^{3}\cdot\left(\tfrac{4}{3}\right)^{4}\\ &\Rightarrow\ =2^{2}\cdot\frac{3^{3}}{2^{3}}\cdot\frac{2^{8}}{3^{4}} \quad\text{(}4^4=2^8\text{)}\\ &\Rightarrow\ =\frac{2^{2+8}\cdot 3^{3}}{2^{3}\cdot 3^{4}}=\frac{2^{10}}{2^{3}\cdot 3} \end{aligned}
4

Divide numerator by denominator.

21021023⋅3=23⋅3=8×3=24\begin{aligned} &\frac{2^{10}}{\dfrac{2^{10}}{2^{3}\cdot 3}}=2^{3}\cdot 3=8\times 3=24 \end{aligned}
Value=24\text{Value}=24

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CAT 2020 Slot 3 QA Q19 — Logarithms solution | TheCATExam