https://file.aiquickdraw.com/mathbot/file/1725808773830ttwsykw5.jpg

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.

Simplify the expression: 2log16 - 3log(1/27) + 5log1

Learn how to simplify the logarithmic expression 2log16 - 3log(1/27) + 5log1 step by step, using key logarithmic properties such as the power rule and logarithmic properties. Suitable for students in Grades 9-12.

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