https://file.aiquickdraw.com/mathbot/file/17274148431480p8rz1l2.jpg

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.

Simplify the expression log_3(27) - log(1/10) - 3 * log(7).

Learn how to simplify the logarithmic expression log_3(27) - log(1/10) - 3 * log(7) using properties of logarithms. This problem focuses on key concepts like logarithmic identities and algebraic simplifications. Suitable for high school students in Grades 9-12.

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