The given expression in the image is:

$2 \log 16 - 3 \log \left(\frac{1}{27}\right) + 5 \log 1$

Let's break this down step by step:

Step 1: Simplifying $\log 16$

We know that: $16 = 2^4$ So, $\log 16 = \log (2^4) = 4 \log 2$.

Thus: $2 \log 16 = 2 \times 4 \log 2 = 8 \log 2$

Step 2: Simplifying $\log \left(\frac{1}{27}\right)$

We know that: $27 = 3^3$ So, $\log \left(\frac{1}{27}\right) = \log (3^{-3}) = -3 \log 3$.

Thus: $3 \log \left(\frac{1}{27}\right) = 3 \times (-3 \log 3) = -9 \log 3$

Step 3: Simplifying $\log 1$

We know that: $\log 1 = 0$ So: $5 \log 1 = 5 \times 0 = 0$

Final Expression:

Now, putting it all together: $8 \log 2 - 9 \log 3 + 0 = 8 \log 2 - 9 \log 3$

This is the simplified form of the expression. Would you like further details or have any other questions?

Here are five related questions:

1. How does the property $\log(ab) = \log a + \log b$ work?
2. What happens when you take the logarithm of a fraction?
3. Can you explain why $\log 1 = 0$?
4. How can the change of base formula help in simplifying logarithmic expressions?
5. What are some common logarithmic properties used to simplify expressions?

Tip: Always break down complex expressions using logarithmic properties like power rules and fractional log rules to simplify step by step.