5log(2) + 3log(3) - 2

Express as single logarithm 

To express the given expression as a single logarithm, we'll follow these steps:

1. **Apply the logarithm property:**
   - Use the property \( a \log_b(c) = \log_b(c^a) \) to move the coefficients inside the logarithms as exponents.

   \[
   5 \log(2) = \log(2^5)
   \]
   \[
   3 \log(3) = \log(3^3)
   \]

2. **Rewrite the expression:**
   \[
   \log(2^5) + \log(3^3) - 2
   \]
   
   Since \( \log(2^5) + \log(3^3) \) is a sum of logarithms, we can use the property \( \log(a) + \log(b) = \log(a \cdot b) \) to combine them:
   
   \[
   \log(2^5 \cdot 3^3) - 2
   \]

3. **Simplify:**
   \[
   2^5 = 32 \quad \text{and} \quad 3^3 = 27
   \]
   
   So the expression becomes:
   \[
   \log(32 \cdot 27) - 2 = \log(864) - 2
   \]

4. **Final step:**
   We still have \( -2 \) outside the logarithm. To combine this, express \( -2 \) as \( \log(10^2) \), because \( \log(10^2) = 2 \). Thus, we have:
   
   \[
   \log(864) - \log(10^2)
   \]

   Using the property \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \), this simplifies to:
   \[
   \log\left(\frac{864}{10^2}\right) = \log\left(\frac{864}{100}\right)
   \]
   
   Finally:
   \[
   \log(8.64)
   \]

So, the expression simplifies to \( \log(8.64) \).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do logarithmic properties help in simplifying expressions?
2. What is the inverse of logarithmic operations?
3. Can you express \( 7\log(4) + \log(5) - 4 \) as a single logarithm?
4. How would you solve an equation involving \( \log(x) + \log(2x) = 3 \)?
5. What are the different bases of logarithms and how do they affect the result?

**Tip:** When simplifying logarithmic expressions, remember that addition inside the logarithms corresponds to multiplication of arguments, and subtraction corresponds to division.

5log(2) + 3log(3) - 2. Express as single logarithm

Learn how to simplify the logarithmic expression 5log(2) + 3log(3) - 2 into a single logarithm using logarithmic properties. This step-by-step solution covers key logarithmic concepts and formulas, ideal for students in Grades 9-12.

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