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To solve the equation \(1.7 \cdot 2^{0.01t} = 1.2 \cdot 2^{0.08t}\) for \(t\), follow these steps:

1. **Isolate the exponential terms:** Divide both sides by the coefficients (1.7 and 1.2):
   \[
   2^{0.01t} = \frac{1.2}{1.7} \cdot 2^{0.08t}
   \]

2. **Simplify the fraction:** 
   \[
   2^{0.01t} = \frac{1.2}{1.7} \cdot 2^{0.08t}
   \]
   \[
   \frac{1.2}{1.7} = \frac{12}{17} \approx 0.70588
   \]

3. **Rewrite the equation:**
   \[
   2^{0.01t} = 0.70588 \cdot 2^{0.08t}
   \]

4. **Isolate the exponential term:** Divide both sides by \(2^{0.08t}\):
   \[
   \frac{2^{0.01t}}{2^{0.08t}} = 0.70588
   \]
   \[
   2^{0.01t - 0.08t} = 0.70588
   \]

5. **Simplify the exponent:**
   \[
   2^{-0.07t} = 0.70588
   \]

6. **Take the natural logarithm of both sides to solve for \(t\):**
   \[
   \ln(2^{-0.07t}) = \ln(0.70588)
   \]
   \[
   -0.07t \ln(2) = \ln(0.70588)
   \]

7. **Solve for \(t\):**
   \[
   t = \frac{\ln(0.70588)}{-0.07 \ln(2)}
   \]

8. **Calculate the numerical value:**

   - \(\ln(0.70588) \approx -0.347073647\)
   - \(\ln(2) \approx 0.693147181\)

   \[
   t = \frac{-0.347073647}{-0.07 \times 0.693147181}
   \]

   \[
   t \approx \frac{-0.347073647}{-0.04852030367}
   \]

   \[
   t \approx 7.154
   \]

So, the value of \(t\) is approximately \(7.154\).

Would you like more details or have any questions?

Here are 8 related questions for further exploration:

1. How do you solve exponential equations using logarithms?
2. What is the natural logarithm and how is it used in solving equations?
3. How do you handle equations with different bases on each side?
4. Can you provide another example of solving an exponential equation?
5. How does the base of the exponential function affect the solution?
6. What are common mistakes to avoid when solving exponential equations?
7. How can you verify the solution of an exponential equation?
8. How do you apply these techniques to real-world problems involving exponential growth or decay?

**Tip:** When solving exponential equations, always consider simplifying the coefficients and isolating the exponential terms to make the equation easier to handle.

Learn how to solve the exponential equation 1.7 * 2^(0.01t) = 1.2 * 2^(0.08t) step-by-step with detailed explanations. This problem involves exponential equations and logarithms, focusing on isolating exponential terms and applying logarithmic functions. Suitable for students in Grades 10-12.

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