A car depreciates at a rate of 15% of its value annually. Carl purchased a new car for $25,000 and wants to know how many years it
will take until the car is worth a specific amount.
a) Formulate a logarithmic function that would find the number of years t it will take for the car to be worth x dollars.
Then
b) How many years will Carl have until he new car is worth about $5000?

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manhartbilly manhartbilly · Answer 1 · 2021-02-24T20:29:34+01:00

Answer:

(a) (log X - log 25000)/log 0.85 = t

(b) about 9.9 years

Step-by-step explanation:

(a) Let X = value and t = elapsed years since purchase; X in dollars

X = 25000 x 0.85 x 0.85 x 0.85 ... with a factor of 0.85 for every year

(If you lose 15% of value, you have 85% left, right?)

So if you do it for t years, you have t as the exponent on the 0.85 factor.

[tex]X = 25000(0.85)^t[/tex] is a good equation, but not logarithmic like they ask.

[tex]log X = log 25000 + log [(0.85)^t][/tex] because the log of a product is the sum of the logs of the factors

log X = log 25000 + t*log 0.85 because the log of a power is power times log

log X - log 25000 = t* log 0.85 just algebra, subtracting from both sides

(log X - log 25000)/log 0.85 = t just algebra, dividing both sides

(b) If we let X = $5000

t = (log 5000 - log 25000)/log 0.85

= (3.69897 - 4.39794)/-0.07058

= 9.90

So t = 9.9 years is a good estimate