Question

If \( \frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} + \frac{1}{\log_5 x} + \frac{1}{\log_6 x} = 1 \), then the value of \( x \) is

• A. 18
• B. 36
• C. 120
• D. 360
• E. 720

Hint:

Use properties of logarithms to simplify and solve logarithmic equations.

Answer 1

Answer: (E)

Step 1: Use the property of logarithms \( \frac{1}{\log_b x} = \log_x b \). 
This simplifies the given equation to: \[ \log_x 2 + \log_x 3 + \log_x 4 + \log_x 5 + \log_x 6 = 1 \] Simplifying the sum gives: \[ \log_x (2 \times 3 \times 4 \times 5 \times 6) = \log_x 720 \] Thus, \( x = 720 \).