Chapter 5: Q 99. (page 309)

Find the value of $\log_{2} 3 \cdot \log_{3} 4 \cdot \dots \cdot \log_{n} (n + 1) \cdot \log_{n + 1} 2$

Short Answer

Expert verified

The obtained result is 1.

Step by step solution

Step 1. Given information.

Consider the given expression.

$\log_{2} 3 \cdot \log_{3} 4 \cdot \dots \cdot \log_{n} (n + 1) \cdot \log_{n + 1} 2$

Step 2. Calculation.

Apply the base change formula to simplify the expression.

$\begin{array}{rcl} \log_{2} 3 \cdot \log_{3} 4 \cdot \dots \cdot \log_{n} (n + 1) \cdot \log_{n + 1} 2 & = & \frac{\log 3}{\log 2} \cdot \frac{\log 4}{\log 3} \cdot \dots \cdot \frac{\log (n + 1)}{\log n} \cdot \frac{\log 2}{\log (n + 1)} \\ = & \frac{1}{\log 2} \cdot \frac{1}{1} \cdot \dots \cdot \frac{1}{1} \cdot \frac{\log 2}{1} \\ = & \frac{\log 2}{\log 2} \\ = & 1 \end{array}$

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Find the value of $\log_{2} 3 \cdot \log_{3} 4 \cdot \dots \cdot \log_{n} (n + 1) \cdot \log_{n + 1} 2$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

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Find the value of $\log_{2} 3 \cdot \log_{3} 4 \cdot \dots \cdot \log_{n} (n + 1) \cdot \log_{n + 1} 2$