a-if-f-x-log-x-2-solve-the-equation-2-f-x-2-2-f-2x-2-4-logf-x-

Question Number 50080 by F_Nongue last updated on 13/Dec/18

a) i f f (x) = l o g (x + 2), s o l v e t h e e q u a t i o n :

2^{f (x - 2)} \times 2^{f (2 x + 2)} = 4^{l o g f (x)}

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Dec/18

f (x - 2) = l o g (x - 2 + 2) = l o g x

f (2 x + 2) = l o g (2 x + 2 + 2) = l o g (2 x + 4)

l o g f (x) = l o g {l o g (x + 2) + 2}

2^{l o g x} \times 2^{l o g (2 x + 4)} = 2^{2 l o g {l o g (x + 2) + 2}}

l o g x + l o g (2 x + 4) = 2 l o g {l o g (x + 2) + 2}

l o g {x (2 x + 4)} = l o g {l o g (x + 2) + 2}^{2}

2 x^{2} + 4 = {l o g (x + 2) + 2}^{2}

2 x^{2} + 4 = 4 + 4 l o g (x + 2) + {l o g (x + 2)}^{2}

2 x^{2} = 4 l o g (x + 2) + {l o g (x + 2)}^{2}

k^{2} + 4 k - 2 x^{2} = 0 k = l o g (x + 2)

k^{2} + 4 k + 4 = 2 x^{2} + 4

{(k + 2)}^{2} = 2 (x^{2} + 2)

k + 2 = \sqrt{2 (x^{2} + 2)}

l o g (x + 2) + 2 = \sqrt{2 (x^{2} + 2)}

Commented by tanmay.chaudhury50@gmail.com last updated on 13/Dec/18

Commented by Pk1167156@gmail.com last updated on 14/Dec/18

nice sir