https://file.aiquickdraw.com/mathbot/file/1727303190026zerhfrbm.jpg

I see the math problems from the image. The problems seem to be about logarithmic expressions. Let me break them down for you:

1. **Problem A**:  
   \[
   3 \log_5 25 \cdot 5 \log_3 81 = \, ?
   \]
   To solve this, we first simplify the logarithmic terms:
   - \(\log_5 25 = 2\) because \(5^2 = 25\).
   - \(\log_3 81 = 4\) because \(3^4 = 81\).

   Thus, the expression becomes:
   \[
   3 \cdot 2 \cdot 5 \cdot 4 = 120.
   \]

2. **Problem B**:  
   \[
   \log_3 \left( 5 \log_5 125 \right) = \, ?
   \]
   First, simplify \(\log_5 125\):
   - \(\log_5 125 = 3\) because \(5^3 = 125\).

   So the expression becomes:
   \[
   \log_3 (5 \cdot 3) = \log_3 15.
   \]
   The exact value of \(\log_3 15\) cannot be simplified further without a calculator, but it would be a decimal value close to \(2.46\).

3. **Problem C**:  
   \[
   2 \log_3 72 - 2 \log_3 3 + 2 \log_3 4 = \, ?
   \]
   Using the properties of logarithms:
   \[
   2 \log_3 \left( \frac{72 \cdot 4}{3} \right) = 2 \log_3 \left( \frac{288}{3} \right) = 2 \log_3 96.
   \]
   The exact value of \(\log_3 96\) is also a decimal, approximately \(3.35\). So:
   \[
   2 \cdot 3.35 = 6.7.
   \]

Let me know if you'd like more details or further clarifications!

Here are 5 related questions to further explore the concept:
1. How do you apply the change of base formula for logarithms?
2. Can logarithmic identities help simplify expressions like \(\log_3 72 + \log_3 4\)?
3. What is the difference between natural logarithms and common logarithms?
4. How can logarithmic functions be used to solve exponential equations?
5. Can you find the derivative of a logarithmic function like \(\log_3 x\)?

**Tip**: Always remember to use properties of logarithms, such as the product rule and the power rule, to simplify complex logarithmic expressions.

What is the value of the following logarithmic expressions?
A. 3 log₅ 25 • 5 log₃ 81
B. log₃ (5 log₅ 125)
C. 2 log₃ 72 - 2 log₃ 3 + 2 log₃ 4

Learn how to solve logarithmic expressions such as 3 log₅ 25 • 5 log₃ 81, log₃ (5 log₅ 125), and 2 log₃ 72 - 2 log₃ 3 + 2 log₃ 4 using logarithmic properties. This tutorial is suitable for students in grades 9-12 and covers logarithmic and exponential principles.

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