if a virus is found in a body and after 1h there are 80 bacteria in the individual. and after 100mins (1h&40mins) there are 320 bacteria.

find the equation of N(t)=a(b^t) where a is the initial value and b is the growth rate and t is time in hours.

ive tried this many times and i just cant figure it out. ive asked my friends and they dont know either

please help

Plugging in the given numbers, N(1) = a(b^1) = 80 => a*b = 80.

and 100mins = ~1.67 hrs (really 1&2/3 hrs) , so N(1.67) = a(b^1.67) = 320.

If I solve the first equation for a, I get a = 80/b. Substituting that into equation 2 gives me

(80/b)(b^1.67) = 320.

move the 80/b to the other side so that the b^1.67 is by itself. => b^1.67 = 320(b/80) =>b^1.67 = 4b

——–This is where it gets tricky———–

(When in doubt take the log of both sides)

Here we're going to need logarithms, and you will want to use log base b to cancel as much as you can. to denote log base b I will write log{b}.

So, log{b}(b^1.67) = log{b}(4b) => 1.67*log{b}(b) = log{b}(4) + log{b}(b) => 1.67 = log{b}4 + 1

=> log{b}(4) = 2/3 (Note that the real decimal is not 0.67 it is a repeating decimal that can be rewritten as 2/3)

=>log{b}(4) = 2/3 => b^(2/3) = 4 => b = 4^(3/2) => b = 8

Now that we've solved for b, we can plug that into the first equation (a*b=80) to get a.

a(8) = 80 => a=10.

This we can plug into the given formula

8^t).

———DONE———

To check, plug your given information back into your solution.

N(1) = 10(8^1) = 80 (good)

N(1.67) = 10(8^1.67) = 320 (good)

n=ab^t

80=a*b^60 and 320=a*b^100

ln80=lna+60lnb and ln320=lna+100lnb

solve this system for lna and lnb

subtract eq1 from eq2: ln320-ln80= (100-60)lnb or ln4=40lnb and lnb=(ln4)/40

5*eq1: 5ln80=5lna + 300lnb

3*eq2: 3ln320= 3lna + 300lnb

subtract them: 5ln80-3ln320= 2lna; ln[80^5/320^3]=2lna; lna=(1/2)ln(80^5/320^3)

Now since lna=(1/2)ln(80^5/320^3), a=sqrt(80^5/320^3) or 10

And lnb=(ln4)/40 and b=4^(1/40) or 1.035265

So we have n= 10*1.035265^t

This answer can be verified by substituting in values of 60 and 100 for t. It checks out.