SOLUTION: Please help me solve this problem. If log(m)n=log(n)m, m not equal to n, m not equal to 1, n not equal to 1 and m.n>0, evaluate m.n. Please note (m) and (n) are subscripts (bases)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Please help me solve this problem. If log(m)n=log(n)m, m not equal to n, m not equal to 1, n not equal to 1 and m.n>0, evaluate m.n. Please note (m) and (n) are subscripts (bases)      Log On


   

Question 242902: Please help me solve this problem.
If log(m)n=log(n)m, m not equal to n, m not equal to 1, n not equal to 1 and m.n>0, evaluate m.n. Please note (m) and (n) are subscripts (bases)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%28m%2C%28n%29%29=log%28n%2C%28m%29%29 Start with the given equation.


log%2810%2C%28n%29%29%2Flog%2810%2C%28m%29%29=log%28n%2C%28m%29%29 Use the change of base formula on the left side.


log%2810%2C%28n%29%29%2Flog%2810%2C%28m%29%29=log%2810%2C%28m%29%29%2Flog%2810%2C%28n%29%29 Use the change of base formula on the right side.


log%2810%2C%28n%29%29log%2810%2C%28n%29%29=log%2810%2C%28m%29%29log%2810%2C%28m%29%29 Cross multiply


%28log%2810%2C%28n%29%29%29%5E2=%28log%2810%2C%28m%29%29%29%5E2 Condense the terms.


%28log%2810%2C%28n%29%29%29%5E2-%28log%2810%2C%28m%29%29%29%5E2=0 Get everything to one side


Factor the left side using the difference of squares formula


log%2810%2C%28n%29%29%2Blog%2810%2C%28m%29%29=0 or log%2810%2C%28n%29%29-log%2810%2C%28m%29%29=0 Use the zero product property.


log%2810%2C%28n%29%29=-log%2810%2C%28m%29%29 or log%2810%2C%28n%29%29=log%2810%2C%28m%29%29 Get the log%2810%2C%28m%29%29 term to the other side.


log%2810%2C%28n%29%29=log%2810%2C%28m%5E%28-1%29%29%29 or log%2810%2C%28n%29%29=log%2810%2C%28m%29%29 Use the identity y%2Alog%28b%2C%28x%29%29=log%28b%2C%28x%5Ey%29%29 to rewrite the right side (of the first equation)


log%2810%2C%28n%29%29=log%2810%2C%281%2Fm%29%29 or log%2810%2C%28n%29%29=log%2810%2C%28m%29%29 Simplify


n=1%2Fm or n=m Now equate the arguments.


Since we're told that m%3C%3En, this means that the only solution is n=1%2Fm (which is the same as saying m=1%2Fn)


Now multiply n and m to get: mn=%281%2Fn%29%28n%29=n%2Fn=1. So the product of m and n is 1.