SOLUTION: It has been a while since i have done logarithms so how would you go about doing this problem? 4x3^x = 7x5^x I know that you want to solve for x but I am not sure how. T

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: It has been a while since i have done logarithms so how would you go about doing this problem? 4x3^x = 7x5^x I know that you want to solve for x but I am not sure how. T      Log On


   

Question 210783: It has been a while since i have done logarithms so how would you go about doing this problem?
4x3^x = 7x5^x
I know that you want to solve for x but I am not sure how.
Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that the first 'x's on each side are multiplication symbols.


4%2A3%5Ex+=+7%2A5%5Ex Start with the given equation.


log%2810%2C%284%2A3%5Ex%29%29=log%2810%2C%287%2A5%5Ex%29%29 Take the log of both sides.


Break up the log using the identity log%28b%2C%28A%2AB%29%29=log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29


Subtract log%2810%2C%284%29%29 from both sides.


Subtract log%2810%2C%285%5Ex%29%29 from both sides.


Rewrite the logs on the left side using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


Factor out the GCF 'x' on the left side.


x%2A%28log%2810%2C%283%2F5%29%29%29=log%2810%2C%287%2F4%29%29 Combine the logs using the identity log%28b%2C%28A%29%29-log%28b%2C%28B%29%29=log%28b%2C%28A%2FB%29%29


x=log%2810%2C%287%2F4%29%29%2F%28log%2810%2C%283%2F5%29%29%29 Divide both sides by log%2810%2C%283%2F5%29%29 to isolate 'x'


x=-1.0955 Approximate the answer (with a calculator)


So the exact answer is x=log%2810%2C%287%2F4%29%29%2F%28log%2810%2C%283%2F5%29%29%29, which can be simplified to and the approximate answer is x=-1.0955