Having an issue with this problem with an assignment Suppose that an algorithm uses $2n^2+3^n$ bit operations to solve a problem of size $n$. Suppose that your machine
can perform one bit operation in $10^{-9}$ seconds, how long does it take your algorithm to solve a problem of
size given below. Note, if your algorithm takes more than $60$ seconds, answer in minutes. For more than $60$
minutes, answer in hours. For more than $24$ hours, answer in days. For more than $365$ days, answer in years.
For more than 100 years, answer in centuries!
(a) 10
(b) 20
(c) 50
(d) 100
(e) 1000
 A: We deal with $n=50$. So we need to evaluate $2(50)^2+3^{50}$. The $2(50)^2$ part is easy to evaluate, by calculator, or, more simply, in one's head. We get $5000$.
Actually, it doesn't matter what we get, since $2(50)^2$ is utterly negligible in comparison with $3^{50}$. 
To evaluate $3^{50}$, we can use a calculator. But because your calculator may later choke on $3^{1000}$, we do it another way. 
We will use logarithms to the base $10$. So by $\log$ we mean $\log$ to the base $10$. Note that $3=10^{\log 3}$, so $3^{50}=10^{50\log 3}$. 
The calculator gives $50\log 3\approx 23.856$, so $3^{50}\approx (10^{23})(10^{0.856})$. We conclude that 
$$3^{50}\approx 7.18\times 10^{23}.$$
Each operation takes $10^{-9}$ seconds. So the number of seconds used is approximately $7.18\times 10^{14}$. 
Divide by $3600$ to get hours, then by $24$ to get days, then by $365$ to get (roughly) years, then by $100$ to get centuries. A long long time! The point of the problem is to give your calculator some exercise, and to show you how fast the function $3^n$ grows. 
Remark: Actually, we don't absolutely need logs to calculate $3^{1000}$.  My cheap scientific calculator says "E" and refuses to do anything if I tell it to find $3^{1000}$. However, it has no trouble declaring that $3^{125}\approx 4.3667\times 10^{59}$. But $3^{1000}$ is the $8$-th power of this, which is about $(4.3667)^8\times 10^{(8)(59)}$.  
