Hello, can someone help me with this tasks?
1.)
\(\log _2\left(x\right)-\log _4\left(y\right)=0 \)
\(5x^2-y^2=4 \)
2.)
\(64\cdot 9^x-84\cdot 12^x+27\cdot 16^x=0 \)
2)
Solve for x:
64 9^x - 7 12^(x + 1) + 27 16^x = 0
The left hand side factors into a product with two terms:
(16 3^x - 9 4^x) (4 3^x - 3 4^x) = 0
Split into two equations:
16 3^x - 9 4^x = 0 or 4 3^x - 3 4^x = 0
Add 9 4^x to both sides:
16 3^x = 9 4^x or 4 3^x - 3 4^x = 0
9 4^x = 2^(2 x)·3^2:
2^4·3^x = 2^(2 x)·3^2 or 4 3^x - 3 4^x = 0
Equate exponents of 2 and 3 on both sides:
4 = 2 x and x = 2 or 4 3^x - 3 4^x = 0
All equations give x = 2 as the solution:
x = 2 or 4 3^x - 3 4^x = 0
Add 3 4^x to both sides:
x = 2 or 4 3^x = 3 4^x
3 4^x = 2^(2 x)·3^1:
x = 2 or 2^2·3^x = 2^(2 x)·3^1
Equate exponents of 2 and 3 on both sides:
x = 2 or 2 = 2 x and x = 1
All equations give x = 1 as the solution:
x = 2 or x = 1
1)
Log_2(x) - Log_4(y) = 0....................(1)
5x^2 - y^2 =4..................................(2)
x = ± sqrt(y^2 + 4)/sqrt(5) sub this into (1) above.
Solve for y:
log(sqrt(y^2 + 4)/sqrt(5))/log(2) - log(y)/log(4) = 0
Bring log(sqrt(y^2 + 4)/sqrt(5))/log(2) - log(y)/log(4) together using the common denominator log(2) log(4):
(log(4) log(sqrt(y^2 + 4)/sqrt(5)) - log(2) log(y))/(log(2) log(4)) = 0
Multiply both sides by log(2) log(4):
log(4) log(sqrt(y^2 + 4)/sqrt(5)) - log(2) log(y) = 0
log(4) log(sqrt(y^2 + 4)/sqrt(5)) - log(2) log(y) = log(y^(-log(2))) + log(5^(-log(2)) (y^2 + 4)^log(2)) = log(5^(-log(2)) y^(-log(2)) (y^2 + 4)^log(2)):
log(5^(-log(2)) y^(-log(2)) (y^2 + 4)^log(2)) = 0
Cancel logarithms by taking exp of both sides:
5^(-log(2)) y^(-log(2)) (y^2 + 4)^log(2) = 1
Multiply both sides by 5^log(2):
y^(-log(2)) (y^2 + 4)^log(2) = 5^log(2)
Factor out the common power log(2) from the left hand side:
((y^2 + 4)/y)^log(2) = 5^log(2)
Raise both sides to the power of 1/log(2):
(y^2 + 4)/y = 5
Multiply both sides by y:
y^2 + 4 = 5 y
Subtract 5 y from both sides:
y^2 - 5 y + 4 = 0
The left hand side factors into a product with two terms:
(y - 4) (y - 1) = 0
Split into two equations:
y - 4 = 0 or y - 1 = 0
Add 4 to both sides:
y = 4 or y - 1 = 0
Add 1 to both sides:
y = 4 or y = 1 and x =+or- 2 or x=+or- 1
