https://file.aiquickdraw.com/mathbot/file/172958568627232pth3lq.jpg

The problem in the image provides a formula for bacterial growth:

\[
t = 5 \cdot \frac{\log(N/60)}{\log 2}
\]

Where:
- \( t \) is the time in hours.
- \( N \) is the population size after time \( t \).
- The initial population size is 60 bacteria.
- The bacteria divide every 5 hours (hence the multiplier 5).
- You're tasked with finding \( t \) when the population grows to **two million** bacteria, or \( N = 2,000,000 \).

### Let's solve for \( t \):

1. Substitute \( N = 2,000,000 \) into the equation:
   \[
   t = 5 \cdot \frac{\log(2,000,000 / 60)}{\log 2}
   \]
   Simplify the fraction inside the logarithm:
   \[
   2,000,000 / 60 = 33,333.33\overline{3}
   \]
   So the equation becomes:
   \[
   t = 5 \cdot \frac{\log(33,333.33\overline{3})}{\log 2}
   \]

2. Now calculate the logarithms:
   \[
   \log(33,333.33) \approx 4.5229
   \]
   \[
   \log 2 \approx 0.3010
   \]

3. Substitute these values back into the equation:
   \[
   t = 5 \cdot \frac{4.5229}{0.3010} \approx 5 \cdot 15.03
   \]

4. Finally, multiply:
   \[
   t \approx 75.15 \, \text{hours}
   \]

### Answer:
The time required for the colony to grow to two million bacteria is approximately **75.15 hours**.

---

Would you like any further clarifications on this? Here are 5 additional questions related to this problem:

1. How does the division rate of bacteria affect the growth time?
2. What would happen to the growth time if the initial number of bacteria were different?
3. How does the logarithmic function apply to real-world growth scenarios like this one?
4. How would the result change if the bacteria divided every 3 hours instead of 5?
5. Can you apply this model to a situation where the growth rate is not constant?

### Tip:
Always check that the units of the inputs match the units expected by the equation to avoid miscalculations.

A certain strain of bacteria divides every 5 hours. If a colony is started with 60 bacteria, how much time is required for the colony to grow to two million bacteria?

Learn how to calculate the time it takes for a bacterial colony, which divides every 5 hours, to grow from 60 to 2 million bacteria using logarithmic growth formulas. Suitable for high school students in Grades 10-12.

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