Question

The value of a car depreciates exponentially over time. The function
v
=
26
,
500
(
2
)
−
0.18
t
can be used to determine v, the value of the car t years after its initial purchase. Which expression represents the number of years that will elapse before the car has a value of $12,000?


log
(
12
,
000
26
,
500
)
0.18


log
(
26
,
500
12
,
000
)
0.18


log
2
(
12
,
000
26
,
500
)
0.18


l
o
g
2
(
12
,
000
26
,
500
)
−
0.18

Answer 1

Set 12000 = 26500·2^(−0.18t). Divide both sides:
12000/26500 = 2^(−0.18t).
Take log base 2:
log_2(12000/26500) = −0.18 t,
so t = log_2(12000/26500) / (−0.18).

This is choice 4. (Equivalently t = log_2(26500/12000) / 0.18.)