Chapter 5: Q 97. (page 309)

Find the value of $\log_{2} 3 \cdot \log_{3} 4 \cdot \log_{4} 5 \cdot \log_{5} 6 \cdot \log_{6} 7 \cdot \log_{7} 8$ .

Short Answer

Expert verified

The obtained result is 3.

Step by step solution

Step 1. Given information.

Consider the given expression.

$\log_{2} 3 \cdot \log_{3} 4 \cdot \log_{4} 5 \cdot \log_{5} 6 \cdot \log_{6} 7 \cdot \log_{7} 8$

Step 2. Calculation.

Apply the base change formula to simplify the expression.

$\begin{array}{rcl} \log_{2} 3 \cdot \log_{3} 4 \cdot \log_{4} 5 \cdot \log_{5} 6 \cdot \log_{6} 7 \cdot \log_{7} 8 & = & \frac{\log 3}{\log 2} \cdot \frac{\log 4}{\log 3} \cdot \frac{\log 5}{\log 4} \cdot \frac{\log 6}{\log 5} \cdot \frac{\log 7}{\log 6} \cdot \frac{\log 8}{\log 7} \\ = & \frac{1}{\log 2} \cdot \frac{1}{1} \cdot \frac{1}{1} \cdot \frac{1}{1} \cdot \frac{1}{1} \cdot \frac{\log 2^{3}}{1} \\ = & \frac{3 \log 2}{\log 2} \\ = & 3 \end{array}$

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Find the value of $\log_{2} 3 \cdot \log_{3} 4 \cdot \log_{4} 5 \cdot \log_{5} 6 \cdot \log_{6} 7 \cdot \log_{7} 8$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

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Find the value of $\log_{2} 3 \cdot \log_{3} 4 \cdot \log_{4} 5 \cdot \log_{5} 6 \cdot \log_{6} 7 \cdot \log_{7} 8$ .