An initial number of bacteria presented in a culture is 10000. This number doubles every 30 minutes. How long will it take to bacteria to reach the number 100000 ?

Explanation

To solve the problem step by step, we will use the information given about the bacteria growth and logarithmic properties.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
- Initial number of bacteria, \( N_0 = 10,000 \)
- Target number of bacteria, \( N = 100,000 \)
- Doubling time, \( t_d = 30 \) minutes

2. **Set up the equation for bacterial growth**:
The number of bacteria after \( t \) minutes can be expressed as:
\[ N = N_0 \times 2^{\frac{t}{t_d}} \]
Plugging in the known values:
\[ 100,000 = 10,000 \times 2^{\frac{t}{30}} \]

3. **Simplify the equation**:
Divide both sides by 10,000:
\[ 10 = 2^{\frac{t}{30}} \]

4. **Take logarithm on both sides**:
We can use the logarithm base 2 for convenience:
\[ \log_2(10) = \frac{t}{30} \]

5. **Solve for \( t \)**:
Multiply both sides by 30:
\[ t = 30 \times \log_2(10) \]

6. **Calculate \( \log_2(10) \)**:
We can use the change of base formula:
\[ \log_2(10) = \frac{\log_{10}(10)}{\log_{10}(2)} = \frac{1}{\log_{10}(2)} \]
The approximate value of \( \log_{10}(2) \) is about \( 0.301 \), so:
\[ \log_2(10) \approx \frac{1}{0.301} \approx 3.32 \]

7. **Substitute back to find \( t \)**:
\[ t \approx 30 \times 3.32 \approx 99.6 \text{ minutes} \]

8. **Round the answer**:
Since we typically round to the nearest whole number, we can say:
\[ t \approx 100 \text{ minutes} \]

### Final Answer:
It will take approximately **100 minutes** for the bacteria to reach the number of 100,000.