using log solve A recent survey conducted in 2023 by the census bureau etermined that the average growth rate for a small town with a population of 35,250 in the northeast united states was approximately 0.65%. what is the doubling time of the population?

To find the doubling time of the population using logarithms, we can use the **Rule of 70** or directly apply the formula for exponential growth. Here's how you can approach it:

### Formula for Doubling Time:
Doubling time \( T \) for an exponential growth process is given by:

\[
T = \frac{\log(2)}{\log(1 + r)}
\]

Where:
- \( r \) is the growth rate (in decimal form).
- \( \log \) refers to the natural logarithm (logarithm base 10 or base \( e \)).

### Given Data:
- The growth rate \( r = 0.65\% = 0.0065 \) (converted to decimal).

### Step-by-step solution:
1. Substitute the values into the formula:

\[
T = \frac{\log(2)}{\log(1 + 0.0065)}
\]

2. Now calculate \( 1 + 0.0065 = 1.0065 \).

\[
T = \frac{\log(2)}{\log(1.0065)}
\]

3. Calculate the logarithms:
   - \( \log(2) \approx 0.3010 \)
   - \( \log(1.0065) \approx 0.0028 \)

4. Finally, divide the two values:

\[
T = \frac{0.3010}{0.0028} \approx 107.5
\]

### Conclusion:
The doubling time for the population is approximately **107.5 years**.

Would you like further clarification on any step?

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Here are 5 related questions you might consider:
1. How would the doubling time change if the growth rate increased to 1%?
2. What is the population expected to be after 50 years given this growth rate?
3. How is the Rule of 70 related to the formula used here?
4. How can logarithmic functions help in modeling population growth over time?
5. Can we use the same formula for populations with negative growth?

**Tip:** The Rule of 70 provides a quick estimate for doubling time using the formula \( T = \frac{70}{r} \), where \( r \) is the percentage growth rate.

A recent survey conducted in 2023 by the census bureau determined that the average growth rate for a small town with a population of 35,250 in the northeast United States was approximately 0.65%. What is the doubling time of the population?

Find out how to calculate the doubling time for a population growing at an average rate of 0.65% using logarithmic functions and exponential growth models. This step-by-step solution demonstrates key concepts such as the doubling time formula, the Rule of 70, and logarithmic properties. Suitable for high school students in Grades 9-12.

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