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The problem you provided asks to estimate the solutions of the following exponential equations:

1. \( 12 (1.32)^t = 25 \)
2. \( 4 (0.093)^t = 10 \)

We'll solve both equations step by step.

### Equation 1: \( 12 (1.32)^t = 25 \)
1. First, divide both sides by 12 to isolate the exponential term:
   \[
   (1.32)^t = \frac{25}{12} \approx 2.083
   \]
2. Take the natural logarithm (ln) of both sides to solve for \( t \):
   \[
   \ln((1.32)^t) = \ln(2.083)
   \]
3. Using the logarithm property \( \ln(a^b) = b \ln(a) \):
   \[
   t \cdot \ln(1.32) = \ln(2.083)
   \]
4. Calculate the natural logarithms:
   \[
   t \cdot 0.27763 \approx 0.73357
   \]
5. Finally, solve for \( t \):
   \[
   t = \frac{0.73357}{0.27763} \approx 2.642
   \]

### Equation 2: \( 4 (0.093)^t = 10 \)
1. Divide both sides by 4:
   \[
   (0.093)^t = \frac{10}{4} = 2.5
   \]
2. Take the natural logarithm of both sides:
   \[
   \ln((0.093)^t) = \ln(2.5)
   \]
3. Using the logarithmic property \( \ln(a^b) = b \ln(a) \):
   \[
   t \cdot \ln(0.093) = \ln(2.5)
   \]
4. Calculate the logarithms:
   \[
   t \cdot (-2.375) = 0.916
   \]
5. Solve for \( t \):
   \[
   t = \frac{0.916}{-2.375} \approx -0.386
   \]

### Final Answers:
1. \( t \approx 2.642 \)
2. \( t \approx -0.386 \)

Would you like any additional details or clarification?

Here are 5 related questions:
1. How do you solve exponential equations when the base isn't easily factorizable?
2. What are some real-life applications of exponential growth and decay?
3. Why is the natural logarithm often used in solving exponential equations?
4. How does the value of the base affect the behavior of an exponential function?
5. What is the inverse of an exponential function, and how is it used in mathematics?

**Tip:** When dealing with exponential equations, logarithms are key for solving for the exponent when the base and the result are known.

Using a calculator, estimate the solution to each of the following equations: 
1. 12(1.32)^t = 25
2. 4(0.093)^t = 10. 
Round your answer to three decimal places.

Learn how to solve exponential equations like 12(1.32)^t = 25 and 4(0.093)^t = 10 using logarithms. This step-by-step guide includes calculations for natural logarithms and key algebraic principles, suitable for high school students.

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