Often new technology spreads exponentially. Between 1995 and 2005, each year the number of Internet domain hosts was 1.43 times the number of hosts in

(a) In light of the fact that, between 1995 and 2005, the total number of hosts for Internet domains increased annually by a factor of 1.43.
This implies that if $N_0$ , followed by the number of hosts in the next year. $N_1$ in the next year is given by $N_1=1.43N_0$ 
Using the number of hosts $N_1$ as the number in the previous year, the number of hosts in the next year $N_2$ is given by $N_2=1.43N_1=1.43(1.43N_0)=(1.43{)}^2{N}_0$ 
Using the number of hosts $N_2$ as the number in the previous year, the number of hosts in the next year $N_3$ is given by $N_3=1.43N_2=1.43((1.43{)}^2{N}_0)=(1.43{)}^3{N}_0$ 
Generalising the above equations, we get $N_t=(1.43{)}^t{N}_0$ 
If t years have passed since 1995, let N be the total number of hosts.
Then, we can replace $N_t$ by N in $N_t=(1.43{)}^t{N}_0$ 
This gives the equation $N=(1.43{)}^t{N}_0$ 
The generic form of an exponential function is comparable to the aforementioned equation.$y=a{b}^x$ where $y\approx N,x\approx t,N_0\approx a,b\approx 1.43$ 
So, the number of hosts is an exponential function of time. 
The number of hosts increases by a factor of 1.43 per year; this growth is exponential because it occurs in steps of 1.43.
(b) As a function of time t and years since 1995, let N represent the number of hosts in millions.
As seen in part (a), we have the equation $N=(1.43{)}^t{N}_0$ where $N_0$ represents the initial number of hosts. 
given that there are 8.2 million hosts at the beginning.
Substitute the value $N_0=8.2$ in $N=(1.43{)}^t{N}_0$ and obtain $N=(1.43{)}^t\times 8.2=8.2(1.43{)}^t$ 
Therefore, the formula for the number of hosts in millions as a function t in years since 1995 is $N=8.2(1.43{)}^t$ 
d) Change the formula's value of N to 49, $N=8.2(1.43{)}^t$ to identify the year in which the number of hosts reached 49 million. 
$49=8.2(1.43{)}^t$ 
$\Rightarrow (1.43{)}^t=\frac{49}{8.2}$ 
$\Rightarrow \ln (1.43{)}^t=\ln (\frac{49}{8.2})$ 
$\Rightarrow t\ln (1.43)=\ln (\frac{49}{8.2})$ 
$\Rightarrow t=\frac{\ln (\frac{49}{8.2})}{\ln (1.43)}$ 
$\Rightarrow t\approx 4.99$ 
$\Rightarrow t\approx 5$ 
Therefore, since 1995, it has taken 5 years for there to be 49 million hosts.
As a result, 49 million hosts were present in the year 2000.