Question

Solve the following systems of equations.
$2^x.x^{-y}=2\sqrt{2},lo{g}_9\frac{1}{x}+0.5=\frac{1}{2}lo{g}_3\left(9y\right).$

Answer 1

Solution:Given,

=> log₉(1/x) + 0.5 = 1/2 log₃9y

=> log₉(1/x) + 1/2 = log₃² (9y) [• 0.5 = 1/2 and 1/n•logₐb = logₐⁿb ]

=> log₉(1/x) + 1/2•log₉9 = log₉(9y) [ • logₐa = 1 ]

=> log₉(1/x)•(3) = log₉(9y)

=> (1/x)• (3) = 9y [• loga = logb, a = b ]

=> x = 1/(3y) equation → (1)

And , 2ˣ•x⁻ʸ = 2√2

=> x⁻ʸ = 2•√(2)/2ˣ

=> x⁻ʸ = 2³⁻²ˣ/₂

Taking log both side having base 2 ,we have

=> log₂x⁻ʸ = (3-2x)/2log₂2 [• logaⁿ = nloga]

=> - ylog₂x = ( 3 - 2x)/2log₂2

=> - ylog₂1/3y = (3 - 2x)/2log₂2 [• value of x from equation (1)]

=> - y•1/3y = (3 - 2x)•2/2 [• logₐb = logₐc , b= c ]

=> - 1/3 = 3 - 2x

=> x = 5/3 equation → (2)

And,From equation (1),we have

=> x = 1/3y

=> y = 1/3x

=> y = 1/(5/3)•3 [• value of x from equation (2) ]

=> y = 1/5

Therefore, solution of the systems of equations are

x = 5/3 , y = 1/5