What is the value of the following expression? log_2 80log_402-log_2 160log_202

2021 AMC 12B Spring Problem 9

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What is the value of the following expression?

\dfrac{\log_2 80}{\log_{40}2}-\dfrac{\log_2 160}{\log_{20}2}

0

1

\dfrac{5}{4}

2

\log_2 5

\dfrac{1}{\log_{40}2}=\log_2 40,

and similarly for the other reciprocal

Write everything in terms of

t=\log_2 5,

since

\log_2 80=4+t,

etc.

Using

\dfrac{1}{\log_{40}2}=\log_2 40

and

\dfrac{1}{\log_{20}2}=\log_2 20,

the expression becomes

(\log_2 80)(\log_2 40)

-(\log_2 160)(\log_2 20).

Let

t=\log_2 5.

Then

\log_2 80=4+t,

\log_2 40=3+t,

\log_2 160=5+t,

\log_2 20=2+t.

The value is

(4+t)(3+t)

-(5+t)(2+t)

=(12+7t+t^2)

-(10+7t+t^2)

=2.

Thus, the correct answer is D.