Find-x-48log-a-4-5log-4-a-a-8-

Question Number 41797 by Tawa1 last updated on 12/Aug/18

Find x : {48 \log}_{a} 4 + {5 \log}_{4} a = \frac{a}{8}

Answered by alex041103 last updated on 13/Aug/18

{l o g}_{b} a = \frac{l n a}{l n b} a n d {l o g}_{a} b = \frac{l n b}{l n a}

\Rightarrow {l o g}_{4} a = \frac{1}{{l o g}_{a} 4}

\Rightarrow l e t {l o g}_{4} a = t

\Rightarrow 48 t^{2} + 5 - \frac{a}{8} t = 0

\Rightarrow t_{1, 2} = \frac{\frac{a}{8} \pm \sqrt{\frac{a^{2}}{64} - 960}}{96} = {l o g}_{4} a

\dots . W f u n c t i o n \dots

Commented by Tawa1 last updated on 13/Aug/18

God bless you sir

Commented by MrW3 last updated on 13/Aug/18

w o r k i n g i s n o t c o r r e c t s i r!

s o l u t i o n f o r

{a x}^{2} + b x + c = 0

i s x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} o n l y

w h e n a, b, c a r e c o n s t a n t s, i . e . w h e n

t h e y a r e i n d e p e n t e n t f r o m x!

b u t i n 48 t^{2} + 5 - \frac{a}{8} t = 0 t a n d a a r e d e p e n d e n t

f r o m e a c h o t h e r, s o y o u m a y n o t a p p l y

t h e f o r m u l a f o r q u a d r a t i c e q u a t i o n .

i f I h a v e a n e q n . l i k e t h i s

{(x^{2} - 1)}^{2} + 2 x (x^{2} - 1) + 1 = 0

y o u c a n n o t s o l v e i t l i k e t h i s :

x^{2} - 1 = \frac{- 2 x \pm \sqrt{4 x^{2} - 4}}{2} = - x \pm \sqrt{x^{2} - 1}

Commented by Tawa1 last updated on 15/Aug/18

God bless you sir .

Commented by alex041103 last updated on 18/Aug/18

T h a t i s n o t a p r o b l e m . I f y o u h a v e s e e n

t h e d e r i v a t i o n o f t h e q u a d r a t i c e q u a t i o n,

i t i s m a d e b y d o i n g v a l i d a l g e b r a i c

m a n i p u l a t i o n . I n o u r c a s e, w e a r e

i n f a c t j u s t t r a n s f o r m i n g t h e e q u a t i o n

i n t o s o m e o t h e r e q u a t i o n, w e {a r e n}^{'} t s o l v i n g i t .

W e j u s t u s e t h e q u a d r a t i c e q u a t i o n

t o t r a n s f o r m i t i n t o s o m e o t h e r e q u a t i o n

w h i c h i s m o r e f a m i l i a r \dots