0-5-log-10-x-2-55x-90-log-10-x-36-log-10-2-Find-the-value-s-of-x-and-determine-the-domain-of-x-

Question Number 4457 by love math last updated on 29/Jan/16

0.5 ({l o g}_{10} (x^{2} - 55 x + 90) - {l o g}_{10} (x - 36)) = {l o g}_{10} \sqrt{2}

F i n d t h e v a l u e (s) o f x a n d d e t e r m i n e t h e d o m a i n o f x .

Commented by Yozzii last updated on 29/Jan/16

T h e d o m a i n o f t h i s e q u a t i o n i s e q u i v a l e n t

t o t h e s e t o f v a l u e s o f x s a t i s f y i n g

t h e e q u a t i o n . M r . S o o m r o h a s g i v e n

t h i s t o b e {3, 54} .

Answered by Rasheed Soomro last updated on 29/Jan/16

0.5 ({l o g}_{10} (x^{2} - 55 x + 90) - {l o g}_{10} (x - 36)) = {l o g}_{10} \sqrt{2}

{l o g}_{10} {(x^{2} - 55 x + 90)}^{1 / 2} - {l o g}_{10} {(x - 36)}^{1 / 2} = {l o g}_{10} 2^{1 / 2}

{l o g}_{10} {(\frac{x^{2} - 55 x + 90}{x - 36})}^{1 / 2} = {l o g}_{10} 2^{1 / 2}

{(\frac{x^{2} - 55 x + 90}{x - 36})}^{1 / 2} = 2^{1 / 2}

\frac{x^{2} - 55 x + 90}{x - 36} = 2

x^{2} - 55 x + 90 - 2 x + 72 = 0

x^{2} - 57 x + 162 = 0

x^{2} - 54 x - 3 x + 162 = 0

x (x - 54) - 3 (x - 54) = 0

(x - 54) (x - 3) = 0

x = 3 ∣ x = 54

I f f (x) = 0.5 ({l o g}_{10} (x^{2} - 55 x + 90) - {l o g}_{10} (x - 36)) - {l o g}_{10} \sqrt{2}

F o r r e a l f (x)

x^{2} - 55 x + 90 > 0 \land x - 36 > 0

x - 36 > 0 \Rightarrow x > 36 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . (i)

E q u a t i o n x^{2} - 55 x + 90 = 0 h a s t w o s o l u t i o n s

x = \frac{55 + \sqrt{2665}}{2}, \frac{55 + \sqrt{2665}}{2}

S o i n e q u a l i t y x^{2} - 55 x + 90 > 0 h a s s o l u t i o n s :

x > \frac{55 + \sqrt{2665}}{2} \approx 53.31, x > \frac{55 - \sqrt{2665}}{2} \approx 1.69 \dots \dots . (i i)

Domain is i n t e r s e c t i o n o f (i) & (i i)

{x : x \in R \land x > \frac{55 + \sqrt{2665}}{2} \approx 53.31}