The expression in the image is:

$5 \log 3 \cdot 3 \log 125 + 16 \log 32$

Let's break it down step by step:

Step 1: Simplify $\log 125$

125 can be written as $5^3$, so: $\log 125 = \log (5^3) = 3 \log 5$

Step 2: Simplify $\log 32$

32 can be written as $2^5$, so: $\log 32 = \log (2^5) = 5 \log 2$

Step 3: Substituting back into the equation

Now substitute these simplified values back into the original expression: $5 \log 3 \cdot 3 \log 125 + 16 \log 32 = 5 \log 3 \cdot 3(3 \log 5) + 16(5 \log 2)$

Simplifying further: $= 5 \log 3 \cdot 9 \log 5 + 80 \log 2$ $= 45 \log 3 \log 5 + 80 \log 2$

The final simplified expression would involve further calculation based on logarithmic values of 2, 3, and 5.

Would you like me to calculate the numerical value, or do you have any questions so far?

Here are five related questions to extend your understanding:

1. What are the properties of logarithms that allow for simplification of powers?
2. How does changing the base of the logarithm affect the calculations?
3. What is the relationship between exponents and logarithms when simplifying?
4. Can logarithmic functions have negative values? Under what conditions?
5. How does the product rule of logarithms apply to expressions with multiple logs?

Tip:

When simplifying logarithmic expressions, remember the key properties: product rule, quotient rule, and power rule for logs.