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 triple in size?

Verified step by step guidance

1. Identify the initial population and the function governing the population growth. The initial population is 150 cells, and the function is given by \( p(t) = 150 \cdot 2^{\frac{t}{12}} \).
2. Determine the target population size for the bacteria to triple. Since the initial population is 150, the target population is \( 3 \times 150 = 450 \) cells.
3. Set up the equation to find the time \( t \) when the population reaches 450 cells: \( 450 = 150 \cdot 2^{\frac{t}{12}} \).
4. Divide both sides of the equation by 150 to isolate the exponential term: \( 3 = 2^{\frac{t}{12}} \).
5. Take the logarithm of both sides to solve for \( t \). Use the property of logarithms that \( \log(a^b) = b \cdot \log(a) \): \( \log(3) = \frac{t}{12} \cdot \log(2) \). Solve for \( t \) by multiplying both sides by 12 and dividing by \( \log(2) \).