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Logarithms

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Question : 83 of 85

Marks: +1, -0

Simplify

\log_{2} 3\; \log_{3} 4\; \log_{4} 5\; \log_{5} 6\; \log_{6} 7\; \log_{7} 8

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

Solution:

Concept:

Use the change-of-base formula to telescope the product.

Explanation:

Write each logarithm using base

e

:

\log_{2}3 = \frac{\ln 3}{\ln 2}

,

\log_{3}4 = \frac{\ln 4}{\ln 3}

,

\ldots

,

\log_{7}8 = \frac{\ln 8}{\ln 7}

.

Multiply:

\frac{\ln 3}{\ln 2} \cdot \frac{\ln 4}{\ln 3} \cdot \frac{\ln 5}{\ln 4} \cdot \frac{\ln 6}{\ln 5} \cdot \frac{\ln 7}{\ln 6} \cdot \frac{\ln 8}{\ln 7} = \frac{\ln 8}{\ln 2}

.

Simplify:

\frac{\ln 8}{\ln 2} = \log_{2}8 = 3

.

Answer:

3

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