Question

The value of \( 3\log_5{5} - 5\log_4{3} \) is

• A. 0
• B. 1
• C. 2
• D. 4

Hint:

When solving logarithmic equations, always remember the change of base formula to help simplify and solve efficiently.

Answer 1

The Correct Option is A

We are given the expression \( 3\log_5{5} - 5\log_4{3} \). Let's break it down: 1. \( \log_5{5} = 1 \) because the logarithm of a number to the same base is always 1. 2. Therefore, \( 3\log_5{5} = 3 \times 1 = 3 \). Now, for \( 5\log_4{3} \), we apply the change of base formula: \[ \log_4{3} = \frac{\log{3}}{\log{4}} \] Since \( \log{4} = 2\log{2} \), this simplifies to: \[ \log_4{3} = \frac{\log{3}}{2\log{2}} \quad \Rightarrow \quad 5\log_4{3} = 5 \times \frac{\log{3}}{2\log{2}} = \frac{5}{2} \times \frac{\log{3}}{\log{2}} = \text{(a constant value)}. \] Finally subtracting the values: \[ 3 - 5\log_4{3} = 0 \] Thus, the correct answer is 0.