Answer
The bladder infection will occur about $4\ \text{hr}$ after 11:00 A.M., or 3:00 P.M.
Work Step by Step
First, find the time $t$ when the bladder contains 25,000 E. coli bacteria, so put $N\left( t \right)=25,000$ in the above expression as,
$\begin{align}
& N\left( t \right)=3000{{\left( 2 \right)}^{\frac{t}{20}}} \\
& 25000=3000{{\left( 2 \right)}^{\frac{t}{20}}} \\
& \frac{25000}{3000}={{\left( 2 \right)}^{\frac{t}{20}}} \\
& 8.33={{\left( 2 \right)}^{\frac{t}{20}}}
\end{align}$
Taking the logarithm on both sides,
$\begin{align}
& \ln \left( 8.33 \right)=\ln {{\left( 2 \right)}^{\frac{t}{20}}} \\
& 2.11=\frac{t}{20}\cdot \ln 2 \\
& 42.2=t\cdot 0.693 \\
& 61.2\approx t
\end{align}$
Now, again find the value of time $t$ when the bladder contains 100,000,000 E. coli bacteria, so put $N\left( t \right)=100,000,000$ in the above expression as,
$\begin{align}
& N\left( t \right)=3000{{\left( 2 \right)}^{\frac{t}{20}}} \\
& 100,000,000=3000{{\left( 2 \right)}^{\frac{t}{20}}} \\
& \frac{100,000,000}{3000}={{\left( 2 \right)}^{\frac{t}{20}}} \\
& 33333.333={{\left( 2 \right)}^{\frac{t}{20}}}
\end{align}$
Taking logarithm on both sides,
$\begin{align}
& \ln \left( 33333.333 \right)=\ln {{\left( 2 \right)}^{\frac{t}{20}}} \\
& 10.414=\frac{t}{20}\cdot \ln 2 \\
& 208.28=t\cdot 0.693 \\
& 300.5\approx t
\end{align}$
So, the time taken to reach 100,000,000 bacteria is,
$300.5\ \text{min}-61.2\text{min}=239.3\text{min}\approx \text{240min}=4\text{hr}$
