Given that x squared + y squared = 5xy,show that; (a) 2log(x+y\√7)=log x + log y. (b). Log ( x- y\√3) = 1\2( log x + log y)
absolutely wrong
x^2 + y^2 = 5xy
complet the square:
x^2 + 2xy + y^2 = 7xy
(x+y)^2 = 7xy
take log of both sides
2log(x+y) = log 7 + logx + logy
2log(x+y) - log7 = logx + logy
2log(x+y) - 2log 7^(1/2) = logx + logy
2[ log(x+y) - log √7] = logx + logy
2 log( (x+y)/√7 ) = logx + logx
as required
to prove the second one, I suggest:
x^2 - 2xy + y^2 = 3xy
and follow the same steps as above
correctl
Thanks you bro
To prove these equations, we'll start with the given equation x^2 + y^2 = 5xy, and then manipulate it step by step to get the desired equations.
(a) To prove 2*log(x + y√7) = log(x) + log(y):
Step 1: Take the logarithm of both sides of the given equation: log(x^2 + y^2) = log(5xy).
Step 2: Using logarithm properties, we can rewrite the equation as log(x^2) + log(y^2) = log(5xy).
Step 3: Using the rules of logarithms, we can then rewrite the equation as 2*log(x) + 2*log(y) = log(5) + log(x) + log(y).
Step 4: Simplify the right side of the equation: log(5) + log(x) + log(y) = log(5xy).
Step 5: Now we have 2*log(x) + 2*log(y) = log(5xy).
Step 6: Divide both sides of the equation by 2: log(x) + log(y) = 1/2 * log(5xy).
Step 7: Using the properties of logarithms, we can simplify the right side of the equation: 1/2 * log(5xy) = log(√(5xy)).
Step 8: Simplify the left side: log(x) + log(y) = log(√(5xy)).
Step 9: Since √(5xy) = x + y√7, we can rewrite it as: log(x) + log(y) = log(x + y√7).
And thus, we have proven that 2*log(x + y√7) = log(x) + log(y).
(b) To prove log(x - y√3) = 1/2 * (log(x) + log(y)):
Step 1: Take the logarithm of both sides of the given equation: log(x^2 + y^2) = log(5xy).
Step 2: Using logarithm properties, we can rewrite the equation as log(x^2) + log(y^2) = log(5xy).
Step 3: Using the rules of logarithms, we can then rewrite the equation as 2*log(x) + 2*log(y) = log(5) + log(x) + log(y).
Step 4: Simplify the right side of the equation: log(5) + log(x) + log(y) = log(5xy).
Step 5: Now we have 2*log(x) + 2*log(y) = log(5xy).
Step 6: Divide both sides of the equation by 2: log(x) + log(y) = 1/2 * log(5xy).
Step 7: Using the properties of logarithms, we can simplify the right side of the equation: 1/2 * log(5xy) = log(√(5xy)).
Step 8: Since √(5xy) = x + y√7, we can rewrite it as: log(x) + log(y) = log(x + y√7).
Step 9: Rearrange the equation to isolate log(x - y√3): log(x + y√7) = log(x) + log(y).
Step 10: Rewrite the equation as: log(x + y√7) = log(x) + log(y).
Step 11: Subtract log(y) from both sides of the equation: log(x + y√7) - log(y) = log(x).
Step 12: Apply the logarithm subtraction property, log(a) - log(b) = log(a/b): log((x + y√7)/y) = log(x).
Step 13: Simplify the left side of the equation: log((x - y√3)) = log(x).
And thus, we have proven that log(x - y√3) = 1/2 * (log(x) + log(y)).
Note: In both cases, we assumed that x, y, x + y√7, and x - y√3 are all positive, which is a common assumption when dealing with logarithms.