Using logarithmic to solve for equation

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 populatio?

Answer 1

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

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

 

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

0.4(2)^t/3 = (0.25)^t/5

ln (0.4(2)^t/3) = ln (0.25)^t/5

ln 0.4 + ln (2)^t/3 = (t/5) ln (0.25)

ln 0.4 + (t/3) ln 2 = (t/5) ln (0.25)

-0.916 + (t/3) 0.693 = (t/5) (-1.386)

-0.916 + 0.231t = -0.0277t

-0.916 = -0.2587t

3.540 = t