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Please help me solve this problem:
If xy = 64 and find x and y.
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= . (1)
Well known rule for the base change of logarithms says that = .
Therefore, the equation (1) becomes
+ = . (2)
To solve it, introduce new variable z = . Then the equation (2) becomes
= , or = 0, or = 0.
Use the quadratic formula to find its roots. The roots are = 2 and/or = .
1. = 2 ====> = 2 ====> y = x^2.
Substitute it into the equation xy = 64, and you will get = 64, which implies x = 4.
In this case, the solution of the original system is x = 4, y = 16.
2. = ====> = ====> y = x^(1/2).
Substitute it into the equation xy = 64, and you will get x^(3/2) = 64, which implies = and hence x = 16.
In this case, the solution of the original system is x = 16, y = 4.
Answer. There are two solutions: 1) x=4, y= 16, and 2) x=16, y = 4.
Solved.
For properties of logarithms and solving logarithmic equations 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".