The population of grasshoppers doubles in a particular field every year. Approximately, how many years will it take the population to grow from 2,000 grasshoppers to 1,000,000 or more?

Answer

To solve the problem of how many years it will take for the population of grasshoppers to grow from 2,000 to 1,000,000, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Doubling Population**:
The population of grasshoppers doubles every year. This means that after one year, the population will be 2 times the initial population.

2. **Set Up the Equation**:
Let \( P \) be the population after \( n \) years. We can express this as:
\[ P = 2000 \times 2^n \]
We want to find \( n \) when \( P \) reaches at least 1,000,000:
\[ 2000 \times 2^n \geq 1,000,000 \]

3. **Simplify the Equation**:
Divide both sides of the inequality by 2000:
\[ 2^n \geq \frac{1,000,000}{2000} \]
Calculate the right side:
\[ \frac{1,000,000}{2000} = 500 \]
So, we have:
\[ 2^n \geq 500 \]

4. **Take Logarithm of Both Sides**:
To solve for \( n \), take the logarithm (base 10) of both sides:
\[ \log(2^n) \geq \log(500) \]

5. **Apply Logarithmic Properties**:
Using the property of logarithms that states \( \log(a^b) = b \cdot \log(a) \), we can rewrite the left side:
\[ n \cdot \log(2) \geq \log(500) \]

6. **Calculate the Values**:
We know:
- \( \log(2) \approx 0.301 \)
- \( \log(500) \approx 2.698 \)

Substitute these values into the inequality:
\[ n \cdot 0.301 \geq 2.698 \]

7. **Solve for \( n \)**:
Divide both sides by \( \log(2) \):
\[ n \geq \frac{2.698}{0.301} \approx 8.95 \]

8. **Round Up**:
Since \( n \) must be a whole number (you can't have a fraction of a year in this context), we round up to the nearest whole number:
\[ n \approx 9 \]

### Conclusion:
It will take approximately **9 years** for the population of grasshoppers to grow from 2,000 to 1,000,000 or more.