A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~\text{hr}$, which means its population is governed by the function $p\left(t\right)=150\cdot{2^{\frac{t}{12}}}$, where $t$ is the number of hours after the first observation.
How long does it take the population to reach $10,000$?

Verified step by step guidance

1. Identify the given function for the population of bacteria: \( p(t) = 150 \cdot 2^{\frac{t}{12}} \). This function describes how the population changes over time, where \( t \) is the time in hours.
2. Set the population function equal to 10,000 to find the time \( t \) when the population reaches this number: \( 150 \cdot 2^{\frac{t}{12}} = 10,000 \).
3. Divide both sides of the equation by 150 to isolate the exponential term: \( 2^{\frac{t}{12}} = \frac{10,000}{150} \).
4. Calculate \( \frac{10,000}{150} \) to simplify the equation: \( 2^{\frac{t}{12}} = 66.67 \).
5. Take the logarithm of both sides to solve for \( t \): \( \log(2^{\frac{t}{12}}) = \log(66.67) \). Use the property of logarithms \( \log(a^b) = b \cdot \log(a) \) to rewrite the left side as \( \frac{t}{12} \cdot \log(2) = \log(66.67) \). Solve for \( t \) by multiplying both sides by 12 and dividing by \( \log(2) \).