SOLUTION: find the two positive integers that satisfies the equation: {{{log(5,(x))log(4,(x))+log(5,(x))log(3,(x))}}}{{{""=""}}}{{{log(5,(x))log(4,(x))log(3,(x))-log(3,(x))log(4,(x))}}}

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: find the two positive integers that satisfies the equation: {{{log(5,(x))log(4,(x))+log(5,(x))log(3,(x))}}}{{{""=""}}}{{{log(5,(x))log(4,(x))log(3,(x))-log(3,(x))log(4,(x))}}}      Log On


   

Question 1062933: find the two positive integers that satisfies the equation:
log%285%2C%28x%29%29log%284%2C%28x%29%29%2Blog%285%2C%28x%29%29log%283%2C%28x%29%29%22%22=%22%22
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

log%285%2C%28x%29%29log%284%2C%28x%29%29%2Blog%285%2C%28x%29%29log%283%2C%28x%29%29%22%22=%22%22

Use the change of base formula to change to logs base 10:  log%28B%2C%28A%29%29%22%22=%22%22log%28%28A%29%29%2Flog%28%28B%29%29

%22%22=%22%22

Get 0 on the right:

%22%22=%22%220

Multiply through by LCD = log(3)∙log(4)∙log(5)

%22%22=%22%220

Factor out [log(x)]2

%22%22=%22%220

Set each factor = 0 and solve:

%28log%28%28x%29%29%29%5E2%22%22=%22%22%220%3B%22     log%28%283%29%29%2Blog%28%284%29%29-log%28%28x%29%29%2Blog%28%285%29%29%22%22=%22%220

The first factor set = 0 has solution x = 1.

The second factor set equal to 0, we solve for log(x)

log%28%283%29%29%2Blog%28%284%29%29-log%28%28x%29%29%2Blog%28%285%29%29%22%22=%22%220
log%28%283%29%29%2Blog%28%284%29%29%2Blog%28%285%29%29%22%22=%22%22log%28%28x%29%29

log%28%283%2A4%2A5%29%29%22%22=%22%22log%28%28x%29%29

log%28%2860%29%29%22%22=%22%22log%28%28x%29%29

60%22%22=%22%22x

So the two solutions are x = 1 and x = 60.

Edwin