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.


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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

	$5^{\sqrt{x}} - 5^{x - 7} = 125 - 25 = 5^{3} - 5^{2}$ $5^{\sqrt{x}} = 5^{3} . \Rightarrow x = 9$ $5^{x - 7} = 5^{2} \Rightarrow x = 9$
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