We need to simplify the given expression:

$2 \left(\frac{4}{\log_5 4}\right) + \left(\frac{3}{\log_7 8}\right)$

Step 1: Apply Change of Base Formula

Using the change of base formula:

$\log_b a = \frac{\log a}{\log b}$

we rewrite the logarithmic terms:

$\log_5 4 = \frac{\log 4}{\log 5}, \quad \log_7 8 = \frac{\log 8}{\log 7}$

Step 2: Rewrite the Expression

Substituting these values:

$\frac{4}{\log_5 4} = \frac{4}{\frac{\log 4}{\log 5}} = 4 \times \frac{\log 5}{\log 4} = \frac{4 \log 5}{\log 4}$

$\frac{3}{\log_7 8} = \frac{3}{\frac{\log 8}{\log 7}} = 3 \times \frac{\log 7}{\log 8} = \frac{3 \log 7}{\log 8}$

Step 3: Compute the Final Expression

$2 \times \frac{4 \log 5}{\log 4} + \frac{3 \log 7}{\log 8}$

$\frac{8 \log 5}{\log 4} + \frac{3 \log 7}{\log 8}$

Now, let’s compute the numerical value.

Using common logarithm values:

$\log 4 \approx 0.6021, \quad \log 5 \approx 0.6990, \quad \log 7 \approx 0.8451, \quad \log 8 \approx 0.9031$

$\frac{8 \times 0.6990}{0.6021} + \frac{3 \times 0.8451}{0.9031}$

$\frac{5.592}{0.6021} + \frac{2.5353}{0.9031}$

$9.29 + 2.81 = 12.10$

Final Answer:

$\mathbf{12.10}$

Would you like a more detailed breakdown?

Related Questions:

1. How does the change of base formula work in logarithms?
2. What are some real-life applications of logarithms?
3. How do you simplify expressions involving multiple logarithmic bases?
4. What is the significance of logarithms in computing?
5. How do you solve logarithmic equations?

Tip:
When working with logarithms in calculations, always check if base conversion simplifies the problem!