Question 1187529
the computer is purchased for 750.
it loses 11% of its value each year.
the function  you are using is:
v(t) = a * b^t
a would be the original amount = 750
b would be the rate that it is growing.
if it shrinking, than b will be less than 1.
you are given that it loses 11% of its value each year.
11% is equal to .11.
11 is the rate percent.
.11 is the rate.
rate percent / 100 = rate
rate * 100 = rate percent.
since it is losing 11% of its value then it is growing by a rate of (1 minus .11) = .89 
the formula becomes v(t) = 750 * .89 ^ t


you want to know how many years it will take for it to lose half its value.
replace v(t) with 750/2 to get:
750/2 = 750 * .89 ^ t
if you divide both sides of that equation by 750, you will get:
(750/2)/750 = .89 ^ t
simplify that to get:
375/750 = .89 ^ t
simplify further to get:
.5 = .89^t


to solve for t, take the log of both sides of the equation to get:
log(.5) = log(.89^t)
by one of the laws of logs, log(.89^t) = t * log(.89)
equation becomes:
log(.5) = t * log(.89
solve for t to get:
t = log(.5) / log(.89)
use your calculator to get:
 = 5.948034681.


confirm by replacing t in our original equation with that value to get:
375 = 750 * .89 ^ 5.948034681.
evaluate to get:
375 = 750 * .5 = 375.
this confirms the solution is correct.


your solution is that t = 5.948034681


here's a reference on logarithm arithmetic.


<a href = "https://www.purplemath.com/modules/logrules.htm" target = "_blank">https://www.purplemath.com/modules/logrules.htm</a>


there is a formula that i use that is more informative.
the formula is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
(1 + r) is the growth rate per time period
n is the number of time periods.


your formula was v(t) = a * b ^ t
your a corresponds to my p
your b corresponds to my growth rate.


if you had used my formula, then:
p = 750
r = -.11
1 + r = 1 - .11 = .89
f = 375


the formula would have become 375 = 750 * .89 ^ n
the growth rate is per year
n is in years.
your time periods are therefore in years.


you would have solve for n the same way.
375 = 750 * .89 ^ n becomes:
375/750 = .89 ^ n
you would then take the log of both sides of the equation to get:
log(375/750) = log(.89^n) which would become:
log(375/750) = n * log(.89) which would become:
n = log(375/750) / log(.89) which would get you the same answer.


hopefully this helps.
let me know if you have any questions.


theo