Test Index

Logarithms

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Question : 9 of 85

Marks: +1, -0

If

\log_2 x + \log_4 x + \log_{16} x = \;\frac{21}{4}

x

is

4
8
16
32

Solution:

Concept:

Convert all logarithms to the same base to combine them into a single equation.

Explanation:

Let

t = \log_2 x

.

Then

\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{t}{2}

.

Similarly,

\log_{16} x = \frac{\log_2 x}{\log_2 16} = \frac{t}{4}

.

Add them:

t + \frac{t}{2} + \frac{t}{4} = \frac{7t}{4}

.

Given

\frac{7t}{4} = \frac{21}{4}

, so multiply both sides by 4:

7t = 21

, hence

t = 3

.

Thus

\log_2 x = 3

, so

x = 2^3 = 8

.

Answer:

The correct option is B (8).

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