The expression \( 2\log{2} + 3\log{3} \) can be simplified using the properties of logarithms.

### Step 1: Apply the power rule of logarithms
The power rule states that \( a\log{b} = \log{b^a} \). So, we can rewrite the expression as:

\[
2\log{2} = \log{2^2} = \log{4}
\]
\[
3\log{3} = \log{3^3} = \log{27}
\]

### Step 2: Combine the logarithms
Now, add the two logarithms together:

\[
\log{4} + \log{27}
\]

Using the addition rule of logarithms, which states that \( \log{a} + \log{b} = \log{(a \times b)} \):

\[
\log{4} + \log{27} = \log{(4 \times 27)} = \log{108}
\]

### Final Answer
The simplified expression is:

\[
2\log{2} + 3\log{3} = \log{108}
\]

Would you like more details, or do you have any questions?

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Here are 8 related questions:
1. What is the power rule for logarithms?
2. How do you combine logarithms with different bases?
3. What happens if you add a constant to a logarithmic expression?
4. How would the result change if the logarithms were in different bases?
5. Can you derive the same result using natural logarithms?
6. How can logarithms be used to solve exponential equations?
7. What is the inverse operation of taking a logarithm?
8. How does the multiplication rule of logarithms simplify the calculation of large products?

**Tip:** The properties of logarithms can often turn complex multiplication or exponentiation problems into simpler addition or multiplication tasks.

Explore how to simplify the expression 2log2 + 3log3 step-by-step using logarithm properties. Learn about the power rule and addition rule of logarithms, essential for high school students studying advanced algebra and logarithms.

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