Math Olympiad Question | Solving a System of Equations with Logarithm in 2 Ways

🔴Math Olympiad Question | Solving a System of Equations with Logarithm in 2 Ways

today we are going to solve a system of equations involving logarithms.

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topics covered in this video:
Logarithm
system of equations
substitution method

#Logarithm #SystemOfEquation #SubstitutionMethod

[ingenuintrutschnig]

Like always: excellently done!
Great & thanks for the inspiration!
👏🙏👍🏹🎯
Best greetings from Austria! 🍀

[levskomorovsky]

log Y + log X = log Y + 1/logY = 10/3
x y x x

( log Y)^2 + log Y -10/3 = 0
x x

Then the same solution.

[davidbrisbane]

If we guessed that x and y where positive integers then we'd only try (x, y) = (2, 8), (8, 2) and (4, 4) as the only possible solutions and we'd find (2, 8) and (8, 2) are solutions, but (4, 4) is not.

[rob]

let log x / log y = z
z + 1/z = 10/3
z^2 + 1 = 10 z/3
3z^2 - 10z + 3 = 0
z = 1/3 or z = 3
log x = 3 log y or 3 log x = log y
log x + log y = 4 log y or log x + log y = 4 log x
xy = y^4 or xy = x^4
xy = 16 = 2^4
if y = 2, x = 8
if x = 2, y = 8

[vcvartak]

2nd method is more simple and systematic

[destruidor]

You have complicate the solution….I just apply log basis 2 in the equation xy=16 and then later on I solved a second equation polynomial….it was much simpler….