Question

In a certain area, the bacteria population, $P$, can be modeled by the equation $P=P_0{10}^{kt}$, where $P_0$ is the bacteria population at the beginning of the experiment, t is the time in hours since the beginning of the experiment, and $k$is a positive constant. Find the correct expression which gives the time, $t$, in terms of $k,P,$ and $P_0$.

• A. $t=\frac{\log \left(\frac{P_0}{P}\right)}{k}$
• B. $t=\frac{\log \left(\frac{P}{P_0}\right)}{k}$
• C. $t=\frac{\log (P\times P_0)}{k}$
• D. $t=\log \left(\frac{P}{P_0}\right)$

Answer 1

Answer: (B)

$t=\frac{\log \left(\frac{P}{P_0}\right)}{k}$

$P=P_0{10}^{kt}\frac{P}{P_0}={10}^{kt}\log \left(\frac{P}{P_0}\right)=\log \left({10}^{kt}\right)\log \left(\frac{P}{P_0}\right)=ktlog10\log \left(\frac{P}{P_0}\right)=kt\times 1(log10=1)kt=\log \left(\frac{P}{P_0}\right)t=\frac{\log \left(\frac{P}{P_0}\right)}{k}$