Answer
There will be $3,000,000$ bacteria after approximately $18.19$ hours.
Work Step by Step
The number of bcteria doubles every hour so the growth factor is $2$.
The initial number of bacteria is $10$.
Thus, the function that models the number of bacteria after $x$ hours is: $y=10 \cdot 2^x$
To know when the number of bacteria will reach $3,000,000$, substitute $3,000,000$ to $y$ then solve for $x$:
\begin{align*}
y&=10\cdot 2^x\\\\
3,000,000 &= 10\cdot 2^x\\\\
\frac{3,000,000}{10}&=\frac{10 \cdot 2^x}{10}\\\\
300,000&=2^x
\end{align*}
Take the common logarithm of both sides to obtain:
\begin{align*}
\log{300,000}&=\log{2^x}\\\\
\log{300,000}&= x \log{2} &\text{(Power Property of Logarithms.)}\\\\
\frac{\log{300,000}}{\log{2}}&= \frac{x \log{2}}{\log{2}} &\text{(Divide $\log{2}$ to both sides.)}\\\\
\frac{\log{300,000}}{\log2}&=x\\\\
x&\approx 18.19 &\text{Evaluate using a calculator.}
\end{align*}
Thus, the bacteria will reach $3,000,000$ after around $18.19$ hours.
