If `log_(4)5=a` and `log_(5)6=b` then what is the value of `log_(3)2`

Options

A. (A)`(1)/(2a+1)`
B. (B)`(1)/(2b+1)`
C. (C)`2ab+1`
D. (D)`(1)/(2ab-1)`

Answer

To solve the problem, we need to find the value of \( \log_3 2 \) given that \( \log_4 5 = a \) and \( \log_5 6 = b \).

### Step-by-Step Solution:

1. **Convert the logarithmic expressions using the change of base formula:**
\[ \log_4 5 = \frac{\log 5}{\log 4} = a \quad \text{(1)} \]
\[ \log_5 6 = \frac{\log 6}{\log 5} = b \quad \text{(2)} \]

2. **Express \( \log 4 \) and \( \log 6 \) in terms of \( \log 2 \) and \( \log 3 \):**
\[ \log 4 = \log(2^2) = 2 \log 2 \quad \text{(3)} \]
\[ \log 6 = \log(2 \cdot 3) = \log 2 + \log 3 \quad \text{(4)} \]

3. **Substitute equations (3) and (4) into equations (1) and (2):**
From (1):
\[ a = \frac{\log 5}{2 \log 2} \implies \log 5 = 2a \log 2 \quad \text{(5)} \]
From (2):
\[ b = \frac{\log 6}{\log 5} = \frac{\log 2 + \log 3}{\log 5} \]
Substitute (5) into this equation:
\[ b = \frac{\log 2 + \log 3}{2a \log 2} \]

4. **Rearranging to find \( \log 3 \):**
Multiply both sides by \( 2a \log 2 \):
\[ 2ab \log 2 = \log 2 + \log 3 \]
Rearranging gives:
\[ \log 3 = 2ab \log 2 - \log 2 = (2ab - 1) \log 2 \quad \text{(6)} \]

5. **Now, we need to find \( \log_3 2 \):**
Using the change of base formula:
\[ \log_3 2 = \frac{\log 2}{\log 3} \]
Substitute (6) into this equation:
\[ \log_3 2 = \frac{\log 2}{(2ab - 1) \log 2} \]
This simplifies to:
\[ \log_3 2 = \frac{1}{2ab - 1} \]

### Final Answer:
Thus, the value of \( \log_3 2 \) is:
\[ \log_3 2 = \frac{1}{2ab - 1} \]