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 populatio? 

With explaintion and using log

Answer 1

Springfield
 
f(t) = 200(2)^t/3
 
Shelby
 
g(t) = 500(1/4)^t/5
 
 
These two towns will have the same population when  f(t) = g(t).
 
 
200(2)^t/3  =  500(1/4)^t/5
 
 
Log both sides of equation.
 
Log 200(2)^t/3 = Log500(1/4)^t/5
 
 
Bring the exponents down as the coefficients of the Logs.
 
(t/3)Log200(2) = (t/5)Log500(1/4)
 
(t/3)Log(400) = (t/5)Log(125)
 
 
Subtract (t/5)Log(125) on both sides of equation so we can set the equation equal to zero.

 

(t/3)Log(400) - (t/5)Log(125) = 0

 

(t/3)2.60  -  (t/5)(2.10) = 0

 

0.86t - 0.42t = 0

 

t(0.86 - 0.42) = 0

 

t = 0

 

 

Based on this, they will never have the same population.