Solve using logarithmic

Question

given: populations of 2 towns Springfield: f(t)=200(2)^t/3
shelbyville:g(t)=500(1/4)^t/5
when will springfeikd and shelbyville have the same populatio? What is that population?
With explaintion and using log Please.

Answer 1

f(t) = 200(2)^t/3

g(t) = 500(0.25)^t/5

 

The populations are equal when f(t) = g(t).

 

200(2)^t/3 = 500(0.25)^t/5

ln [200(2)^t/3] = ln [500(0.25)^t/5]

ln 200 + ln 2^t/3 = ln 500 + ln 0.25^t/52

5.298 + (t/3) ln 2 = 6.215 + (t/5) ln 0.25

5.298 + 0.693(t/3) = 6.215 - 1.386(t/5)

5.298 + 0.231t = 6.215 - 0.277t

5.298 + 0.508t = 6.215

0.508t = 0.917

t = 1.805

 

For the population, calculate either

 

f(1.805) = 200(2)^1.805/3    or

g(1.805) = 500(0.25)^1.805/5