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The image shows the following system of equations:

1. \( 9^a = 5 \)
2. \( 25^b = 2 \)
3. \( 8^c = 3 \)

You are asked to find the value of \( a \cdot b \cdot c \).

### Step-by-step approach:

To solve for \( a \), \( b \), and \( c \), we can take the logarithm of both sides of each equation.

1. From \( 9^a = 5 \), taking the natural logarithm (or log base 10) on both sides:
   \[
   \ln(9^a) = \ln(5) \quad \Rightarrow \quad a \ln(9) = \ln(5) \quad \Rightarrow \quad a = \frac{\ln(5)}{\ln(9)}
   \]
   
2. From \( 25^b = 2 \), taking the logarithm on both sides:
   \[
   \ln(25^b) = \ln(2) \quad \Rightarrow \quad b \ln(25) = \ln(2) \quad \Rightarrow \quad b = \frac{\ln(2)}{\ln(25)}
   \]
   
3. From \( 8^c = 3 \), taking the logarithm on both sides:
   \[
   \ln(8^c) = \ln(3) \quad \Rightarrow \quad c \ln(8) = \ln(3) \quad \Rightarrow \quad c = \frac{\ln(3)}{\ln(8)}
   \]

Now, we calculate the values of \( a \), \( b \), and \( c \) using approximate logarithms:

\[
a = \frac{\ln(5)}{\ln(9)} \approx \frac{1.6094}{2.1972} \approx 0.732
\]
\[
b = \frac{\ln(2)}{\ln(25)} \approx \frac{0.6931}{3.2189} \approx 0.2154
\]
\[
c = \frac{\ln(3)}{\ln(8)} \approx \frac{1.0986}{2.0794} \approx 0.5283
\]

Finally, we find \( a \cdot b \cdot c \):

\[
a \cdot b \cdot c \approx 0.732 \cdot 0.2154 \cdot 0.5283 \approx 0.0833
\]

### Answer:
\( a \cdot b \cdot c \approx 0.0833 \).

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do logarithmic properties simplify the solving of exponential equations?
2. Why is the natural logarithm often used in solving exponentiation problems?
3. What is the importance of approximate values in real-life mathematical applications?
4. How would this approach change if different logarithm bases were used (e.g., base 10)?
5. Can this method be applied to solve equations involving roots instead of powers?

**Tip:** When solving exponential equations, taking logarithms is a powerful tool to isolate variables in the exponent.

9^a = 5, 25^b = 2, 8^c = 3; Find a * b * c.

Learn how to solve a system of exponential equations using logarithms, involving expressions like 9^a = 5, 25^b = 2, and 8^c = 3. This solution uses logarithmic properties to simplify and calculate the values of a, b, and c, and determine their product. Suitable for students in Grades 9-12.

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