Question

A population of 200 animals is decreasing by a rate of .96 per year . At this rate, in how many years will the population be less than 170?

Answer 1

At the end of the zeroth year, the population is 200.
At the end of the first year, the population is 200(0.96)¹
At the end of the second year, the population is 200(0.96)²
We can generalise this to become at the end of the nth year as 200(0.96)ⁿ

Now, we need to know when the population will be less than 170.

So, 170 ≤ 200(0.96)ⁿ
170/200 ≤ 0.96ⁿ
17/20 ≤ 0.96ⁿ
Let 17/20 = 0.96ⁿ, first.
log_0.96(17/2) = n

n = ln(17/20)/ln(0.96)
n will be the 4th year, as after the third year, the population reaches ≈176