SOLUTION: if n^2+m^2 = 11nm, show that 1/2(logn +logm) =log(n-m/3).

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Question 1097315: if n^2+m^2 = 11nm, show that
1/2(logn +logm) =log(n-m/3).
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
n%5E2%2Bm%5E2 = 11nm   is equivalent to  

n%5E2-+2mn+%2B+m%5E2 = 9nm   is equivalent to  

%28n-m%29%5E2 = 9nm   is equivalent to  

%28%28n-m%29%5E2%29%2F9 = nm   is equivalent to  

%28%28n-m%29%2F3%29%5E2 = nm.



Under the assumption  n > 0,  m > 0,  n > m  it implies that


2*log((n-m)/3) = log(n) + log(m),    which is equivalent to


(1/2)*(log(n) + log(m)) = log((n-m)/3).

QED.


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On logarithms and their properties, see the lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
    - Solving logarithmic equations
    - Using logarithms to solve real world problems
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Logarithms".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.