3-log-3x-4-4-log-4x-3-

Question Number 1343 by Rasheed Soomro last updated on 24/Jul/15

3^{l o g 3 x + 4} = 4^{l o g 4 x + 3}

Answered by 112358 last updated on 24/Jul/15

T a k i n g l o g s t o b a s e e o n b o t h s i d e s

\Rightarrow (l o g 3 x + 4) l n 3 = (l o g 4 x + 3) l n 4

(l o g x + l o g 3 + 4) l n 3 = (l o g x + l o g 4 + 3) l n 4

l e t p = l o g x

∴ p l n 3 + (l o g 3 + 4) l n 3 = p l n 4 + (l o g 4 + 3) l n 4

p (l n 3 - l n 4) = (l o g 4 + 3) l n 4 - (l o g 3 + 4) l n 3

p = \frac{(l o g 4 + 3) l n 4 - (l o g 3 + 4) l n 3}{l n (3 / 4)}

S i n c e p = l o g x

\Rightarrow l o g x = \frac{(l o g 4 + 3) l n 4 - (l o g 3 + 4) l n 3}{l n (3 / 4)}

∴ x = 10^{\frac{(l o g 4 + 3) l n 4 - (l o g 3 + 4) l n 3}{l n (3 / 4)}}

x \approx 0.549111367

A l t e r n a t i v e l y

P l o t t i n g t h e g r a p h o f

y = 3^{l o g 3 x + 4} - 4^{l o g 4 x + 3} a n d r e a d i n g o f f

i t s r e a l r o o t g i v e s a n a p p r o x i m a t e

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

e q u a t i o n . I f r e a d i n g o f f d o e s n o t

y i e l d a n a c c u r a t e r e s u l t o t h e r

n u m e r i c a l m e t h o d s l i k e N e w t o n -

{R a p h s o n}^{'} s i t e r a t i v e f o r m u l a r,

l i n e a r i n t e r p o l a t i o n o r i n t e r v a l

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

i f w e d e f i n e \exists x [0, 1] ∣ y = 0 .

Commented by Rasheed Ahmad last updated on 24/Jul/15

G o o d w o r k!