The given expression from the image is:

$\log_3 27 - \log \frac{1}{10} - 3 \cdot \log 7$

Let's simplify this step by step:

1. Simplifying $\log_3 27$: $\log_3 27 = \log_3 (3^3) = 3$

2. Simplifying $\log \frac{1}{10}$: Using the logarithmic property $\log \frac{a}{b} = \log a - \log b$, $\log \frac{1}{10} = \log 1 - \log 10$ Since $\log 1 = 0$, $\log \frac{1}{10} = 0 - \log 10 = -1$ (assuming the base of this logarithm is 10).

3. Simplifying $-3 \cdot \log 7$: This term remains as is: $-3 \cdot \log 7$

Now, substituting these values back into the expression:

$3 - (-1) - 3 \cdot \log 7$

This simplifies to:

$3 + 1 - 3 \cdot \log 7 = 4 - 3 \cdot \log 7$

Thus, the simplified expression is:

$4 - 3 \cdot \log 7$

Would you like further details or explanations?

Here are some related questions to consider:

1. What are the properties of logarithms used in simplifying this expression?
2. How does the base of a logarithm affect the result?
3. How would the solution change if the base of the logarithm was different (e.g., natural logarithms)?
4. What real-world applications use logarithms like those shown in this problem?
5. Can you apply similar logarithmic properties to expressions with multiple variables?

Tip: Always check the base of the logarithm before simplifying expressions, as it can change the outcome significantly.