Simplify: -3(ln4/ln5 - log₅7 + 1/log₆5) Logarithmic Expression

$-3(\frac{\ln4}{\ln5}-\log_57+\frac{1}{\log_65})=$

To solve this problem, we'll follow these steps:

• Step 1: Apply the change-of-base formula to $\frac{\ln 4}{\ln 5}$.

• Step 2: Apply the reciprocal property to $\frac{1}{\log_6 5}$.

• Step 3: Use the subtraction property of logs to simplify the expression.

• Step 4: Combine the simplified logarithms and multiply by -3.

Now, let's work through each step:

Step 1: Using the change-of-base formula, we have $\frac{\ln 4}{\ln 5} = \log_5 4$.

Step 2: Apply the reciprocal property to the third term: $\frac{1}{\log_6 5} = \log_5 6$.

Step 3: Substitute into the expression: $-3(\log_5 4 - \log_5 7 + \log_5 6)$.

Step 4: Combine terms using the properties of logs: $\log_5 4 - \log_5 7 + \log_5 6 = \log_5 \left(\frac{4 \times 6}{7}\right)$.

Step 5: Simplify to get: $\log_5 \left(\frac{24}{7}\right)$.

Multiply by -3: $-3(\log_5 (\frac{24}{7})) = 3\log_5 \left(\frac{7}{24}\right)$.

Therefore, the solution to the problem is $3\log_5 \frac{7}{24}$.