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Logarithms

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Question : 13 of 85

Marks: +1, -0

If

\log _{4} x+\log _{8} x+\log _{2} x=11

x

is

8
16
64
128

Solution:

Concept:

Express all logarithms with base 2 using the change of base formula.

Explanation:

Let

t = \log_{2} x

.

Then

\log_{4} x = \frac{\log_{2} x}{\log_{2} 4} = \frac{t}{2}

and

\log_{8} x = \frac{\log_{2} x}{\log_{2} 8} = \frac{t}{3}

.

The equation becomes

\frac{t}{2} + \frac{t}{3} + t = 11

.

Multiply both sides by

6

:

3t + 2t + 6t = 66 \implies 11t = 66 \implies t = 6

.

Thus

\log_{2} x = 6 \implies x = 2^{6} = 64

.

Answer:

64

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