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WAEC Mathematics 2011 Paper

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Question : 12

Total: 47

Simplify

\frac{\log \sqrt{27}}{\log \sqrt{81}}

3
2
$\frac{3}{2}$
$\frac{3}{4}$

Solution:

Concept:

Use logarithm power rules to simplify the ratio of logarithms.

Explanation:

\sqrt{27} = 27^{\frac{1}{2}}

, and

\sqrt{81} = 81^{\frac{1}{2}}

.

Write

\frac{\log \sqrt{27}}{\log \sqrt{81}} = \frac{\log 27^{\frac{1}{2}}}{\log 81^{\frac{1}{2}}}

.

Use the power rule:

\log a^b = b \log a

.

Numerator:

\log 27^{\frac{1}{2}} = \frac{1}{2} \log 27 = \frac{1}{2} \log 3^3 = \frac{3}{2} \log 3

.

Denominator:

\log 81^{\frac{1}{2}} = \frac{1}{2} \log 81 = \frac{1}{2} \log 3^4 = 2 \log 3

.

Therefore, the ratio is

\frac{\frac{3}{2} \log 3}{2 \log 3} = \frac{3}{2} \times \frac{1}{2} = \frac{3}{4}

.

Answer:

D.

\frac{3}{4}

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