help i’m a little lost

Question

A small town currently only has a population of 1000 people. however thanks to the town being featured in a tv show the board of tourism is hoping that the population will double every decade. how long will it take for the population to reach 32000 if they are right.

Answer 1

initial population = 1,000

it doubles in 10 years to 2,000

in 10 more years it doubles again to 4,000. that's 20 years total

then 10 more years, it's 8,000, for 30 years total

10 more, & it's 16,000 for 40 years total

10 more, it's 32,000 for 50 years total

but the natural annual growth rate would be r in an exponential function, where t= number of years and e= approximately 2.71828

doubling time means 10 in the following equation

2 = e^10r

take natural logs of both sides to get

ln2 =10r

r =ln2/10 = .0693 = 6.93 % annual growth rate

32=(1)e^.0693t where t = number of years until 1 thousand reaches 32 thousand

ln32 = .0693t

t =ln32/.0693 = 50 years to reach 32,000 population

General formula is A=Pe^rt. P is the starting amount, A the ending amount, r=annual growth rate, t= years

Answer 2

We can use the exponential growth formula for this, which is:

f(x) = a(1 + r)^x

Where a is the initial amount, r is the rate of change, and x is the time. In this case, the initial amount is 1000 and the rate of change is 1 (100% for doubling). So our equation for this specific case is:

f(x) = 1000 * (1 + 1)^x = 1000 * 2^x

We want to know when this population will grow to 32000, so we set it equal to that:

1000 * 2^x = 32000

Where x is the number of decades. Can you solve for x with the given equation?