Here-s-a-question-that-has-troubled-me-for-months-now-Please-help-5-x-5-x-7-100-find-the-possible-value-s-of-x-

Question Number 41707 by Necxx last updated on 11/Aug/18

{H e r e}^{'} s a q u e s t i o n t h a t h a s

t r o u b l e d m e f o r m o n t h s n o w . P l e a s e

h e l p

5^{\sqrt{x}} - 5^{x - 7} = 100

f i n d t h e p o s s i b l e v a l u e (s) o f x .

Answered by alex041103 last updated on 11/Aug/18

l e t f (x) = 5^{\sqrt{x}} - 5^{x - 7}

W e w a n t f (x) > 0

\Rightarrow 5^{\sqrt{x}} > 5^{x - 7}

\Rightarrow \sqrt{x} > x - 7

l e t u = \sqrt{x}, u > 0

\Rightarrow u^{2} - u - 7 < 0

f o r u \in (\frac{1 - \sqrt{29}}{2}, \frac{1 + \sqrt{29}}{2}) \cup (0, \infty)

\Rightarrow \sqrt{x} \in (0, \frac{1 + \sqrt{29}}{2})

\Rightarrow x \in (0, \frac{15 + \sqrt{29}}{2}) \approx (0, 10.19)

\Rightarrow s o l u t i o n s w i l l b e b e t w e e n 0 a n d 10.19

W e c a n i m m e d i a t l y s e e t h a t

x = 9 i s a s o l u t i o n .

W e s e e t h a t f^{'} (x) = l n (5) (\frac{5^{\sqrt{x}}}{2 \sqrt{x}} - 5^{x - 7}) .

W e s e e t h a t f^{'} (9) < 0 \Rightarrow t h e f u n c t i o n

h a s v a l u e s b i g e r t h a n 100 .

B y i n i t i a l i z i n g n u m e r i c a l m e t h o d s

w e s e e t h a t x \approx 8.73 i s a s o l u t i o n .

B y i n i t i a l i z i n g t h e g r a p h o f f

w e s e e t h a t t h e r e a r e 2 s o l u t i o n s .

A n s . x = 9 a n d x \approx 8.73

Commented by alex041103 last updated on 11/Aug/18

Commented by alex041103 last updated on 11/Aug/18

f o r e x a m p l e - {N e w t o n}^{'} s m e t h o d

Commented by Necxx last updated on 11/Aug/18

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

a p p l y ?

Commented by Necxx last updated on 11/Aug/18

t h a n k s b o s s

Answered by peter frank last updated on 04/Oct/18

5^{\sqrt{x}} - 5^{x - 7} = 125 - 25 = 5^{3} - 5^{2}

5^{\sqrt{x}} = 5^{3} . \Rightarrow x = 9

5^{x - 7} = 5^{2} \Rightarrow x = 9