Question

How many hours will it take a culture of bacteria to increase from 20 to 2000 if the growth rate per hour is 85%

Answer 1

So you are seeing how much time it'll take so you solve for "t", the time. 
So you take the formula A=Pe^(rt) 
A=2000 because it's the end value 
P=20 because it's the starting value 
r=.85 since 85%=.85 and .85 is the rate 
Plug the values in and you get 2000=20e^(.85t) 
What you do is you divide by 20 so you get 100=e^(.85t) 
Take the natural logarithm of both sides 'cause of e and a natural log is written as ln so you get 
ln 100=.85t ln e and because you can use the power rule you end up with .85t ln e and 
ln e=1 so you have ln 100 = .85t so you divide by .85 so (ln 100)/.85=t and t=5.4178472776331 
hours 
3. Exponential decay: 
A= Pe^(rt) 
where 
A is the final amount 
P is the initial value 
r is rate of decay 
t is time (years) 
Let's say x is the initial amount then (1/2)x=xe^(32r) 
I used x because the value isn't given but anyway division by x would give you 1/2=e^(32r) 
Take the ln of both sides so ln 1/2=32r ln e and then ln e=1 so ln 1/2=32r. 
Divide both sides by 32 and you'd get (ln 1/2)/32=r and r= -0.021660849392498 
4. Another depreciation question. 
Each year the item retains 88% of its last-year value. 
Solve: 250,000(0.88)^x = 100,000 
0.88^x = 0.4 
x = [log0.4]/[log0.88] 
x = 7.168 years