Question: Scientists are working on a second antibacterial drug. To test the drug they need $1,000,000$ bacteria. If they have $500$ bacteria, and the bacteria double in number every $30$ minutes, how long will it take the scientists to be able to test the drug?
1. **State the problem:** We start with $500$ bacteria, which double every $30$ minutes. We want to find how long it takes to reach $1,000,000$ bacteria.
2. **Formula used:** The number of bacteria after time $t$ (in minutes) when doubling every $30$ minutes is given by:
$$ N(t) = N_0 \times 2^{\frac{t}{30}} $$
where $N_0 = 500$ is the initial number of bacteria.
3. **Set up the equation:** We want $N(t) = 1,000,000$, so:
$$ 1,000,000 = 500 \times 2^{\frac{t}{30}} $$
4. **Divide both sides by 500:**
$$ \frac{1,000,000}{500} = 2^{\frac{t}{30}} $$
$$ 2,000 = 2^{\frac{t}{30}} $$
5. **Take the logarithm base 2 of both sides:**
$$ \log_2(2,000) = \log_2\left(2^{\frac{t}{30}}\right) $$
$$ \log_2(2,000) = \frac{t}{30} $$
6. **Calculate $\log_2(2,000)$:**
Since $2^{11} = 2048$ and $2^{10} = 1024$, $\log_2(2,000)$ is approximately $10.97$.
7. **Solve for $t$:**
$$ t = 30 \times 10.97 = 329.1 \text{ minutes} $$
8. **Convert minutes to hours:**
$$ 329.1 \text{ minutes} \div 60 = 5.49 \text{ hours} $$
**Final answer:** It will take approximately $329$ minutes or about $5.5$ hours for the bacteria to reach $1,000,000$ and be ready for testing.