log and population word problems!?

Question

1. the annual rate of deprecation, r, on a car that is purchased for P dollars and is worth W dollars t years later can be found from:
log (1-r) = 1/t log w/p

find the annual rate of depreciation on a car that is purchased for $14650 and sold 3 years later for $9500

answer: 13.4 %

2. the population of a city can be estimated by the equation
p= 108000e^0.05t
where t is the number of years fmor the present time, how many yeras will it take the population to reach 324,000?
answer: 22.0 years


i just dont understand how to do it! please help! thank you!

Answer 1

Question 1
If you are to ask questions at this level, the question must be perfectly clear, else valuable time is wasted.
Your question may be read in a number of different ways due to lack of brackets.
I am going TO TAKE A GUESS at:-
log (1 - r) = (1 / t) log (w / p)
log (1 - r) = log [ ( w / p )^(1 / t) ]
1 - r = (w / p)^(1/t)
1 - r = [ 9500 / 14650 ]^(1/t)
r = 1 - [ 9500 / 14650 ]^(1/3)
r = 1 - 0.866
r = 0.134
r = 13.4%

Question 2
p = 108,000 e^(0.05 t)
p / 108,000 = e^(0.05 t)
ln ( p / 108,000) = 0.05 t
ln (324,000 / 108,000) = 0.05 t
(1 / 0.05) ln (324,000 / 108,000) = t
t = 22 years

PS
Remember the brackets!!
Good luck.

Answer 2

substitute all the know values

log(1-r) = 1/3xlog(9500/14650)
1-r = 0.648 ^ (1/3) = 0.866
r = 0.134=13.4%

324000 = 108000e^0.05t
3 = e^0.05t
t=ln 3/0.05=22