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To solve this problem, we need to determine how many years it takes for the number of tigers to decay by a factor of \( \frac{4}{5} \).

The given function is:

\[
N(t) = 650 \cdot \left( \frac{16}{25} \right)^t
\]

This is an exponential decay function where \( \frac{16}{25} \) represents the decay factor per year.

### Step-by-step solution:
1. **Set up the decay equation**:
   We want to find the time \( t \) when the population decays by a factor of \( \frac{4}{5} \). So, set up the equation:

   \[
   \left( \frac{16}{25} \right)^t = \frac{4}{5}
   \]

2. **Take the logarithm of both sides** to solve for \( t \):

   \[
   \log \left( \left( \frac{16}{25} \right)^t \right) = \log \left( \frac{4}{5} \right)
   \]

3. **Simplify using logarithm rules**:

   \[
   t \cdot \log \left( \frac{16}{25} \right) = \log \left( \frac{4}{5} \right)
   \]

4. **Solve for \( t \)**:

   \[
   t = \frac{\log \left( \frac{4}{5} \right)}{\log \left( \frac{16}{25} \right)}
   \]

5. **Calculate the values** using a calculator:

   - \(\log \left( \frac{4}{5} \right) \approx -0.0969\)
   - \(\log \left( \frac{16}{25} \right) \approx -0.1549\)

   So,

   \[
   t \approx \frac{-0.0969}{-0.1549} \approx 0.63
   \]

### Final answer:
The number of tigers decays by a factor of \( \frac{4}{5} \) every **0.63 years**.

Would you like more details on any step or have any questions? 

Here are five related questions to explore further:
1. How do you find the decay factor for a different percentage decrease?
2. What is the half-life of a population given a decay function like this?
3. How would the decay factor change if the base were \( \frac{25}{16} \) instead?
4. How can this function be used to predict the population at a specific time?
5. How does the initial value affect the overall decay of the population?

**Tip**: When dealing with exponential functions, logarithms are a powerful tool to solve for unknown time values.

Ju Wenjun is an ecologist who studies the change in the tiger population of Siberia over time. The relationship between the elapsed time t, in years, since Ju Wenjun started studying the population, and the number of tigers, N(t), is modeled by the following function:

N(t) = 650 * (16/25)^t

Complete the following sentence about the rate of change in the tiger population. Round your answer to two decimal places.

The number of tigers decays by a factor of 4/5 every ___ years.

This step-by-step solution explores the decay in a tiger population using an exponential decay model, N(t) = 650 * (16/25)^t. Learn how logarithms help determine the time it takes for the population to decay by a factor of 4/5. Suitable for students in Grades 10-12.

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