The expression shown in the image is:

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

Let's break this down and simplify it step by step.

1. Simplify $\log 16$: $\log 16 = \log (2^4) = 4 \log 2$ So, $2 \log 16 = 2 \times 4 \log 2 = 8 \log 2$

2. Simplify $\log \left( \frac{1}{27} \right)$: $\log \left( \frac{1}{27} \right) = \log (27^{-1}) = -\log 27$ Since $27 = 3^3$, we can write: $\log 27 = \log (3^3) = 3 \log 3$ So, $-\log 27 = -3 \log 3$ and $3 \log \left( \frac{1}{27} \right) = 3 \times (-3 \log 3) = -9 \log 3$

3. Simplify $5 \log 1$: $\log 1 = 0$ Therefore, $5 \log 1 = 5 \times 0 = 0$

Now, putting everything together: $8 \log 2 - 9 \log 3 + 0$

This is the simplified form of the expression: $8 \log 2 - 9 \log 3$

Would you like more details or have any further questions?

Here are some related questions for practice:

1. How would you simplify $3 \log 8 + 2 \log 5$?
2. What happens when you multiply two logarithms, for example, $\log a \times \log b$?
3. Can you simplify $\log (2 \times 8)$?
4. How does the logarithmic identity $\log (a^b) = b \log a$ work in other examples?
5. How do you simplify $\log \frac{1}{x}$?

Tip: Remember that $\log 1 = 0$, which can often simplify expressions.