find-x-2-x-3-x-13-

Question Number 42019 by Tawa1 last updated on 16/Aug/18

find x : 2^{x} + 3^{x} = 13

Commented by math khazana by abdo last updated on 17/Aug/18

l e t f (x) = 2^{x} + 3^{x} - 13 (e) \Leftrightarrow f (x) = 0

f (x) = e^{x l n (2)} + e^{x l n 3} - 13 \Rightarrow

f^{^{'}} (x) = l n (2) 2^{x} + l n (3) 3^{x} > 0 s o f i s i n c r e a s i n g

o n R w e h a v e {l i m}_{x \to - \infty} f (x) = - 13 a n d

{l i m}_{x \to + \infty} f (x) = + \infty s o \exists x_{0} \in R / f (x_{0}) = 0

w e h a v e f (0) = - 13 < 0 a n d f (3) = 2^{3} + 3^{3} - 13

= 8 + 27 - 13 = 22 > 0 \Rightarrow x_{0} \in] 0, 3 [w e c a n

v e r i f y t h a t x_{0} = 2 i s e x a c t s o l u t i o n t o f (x) = 0

Commented by Tawa1 last updated on 17/Aug/18

God bless you sir

Commented by maxmathsup by imad last updated on 17/Aug/18

t h a n k s s i r

Answered by MJS last updated on 17/Aug/18

x = 2

Commented by Tawa1 last updated on 17/Aug/18

yes sir . but any workings ? ?

Commented by MJS last updated on 17/Aug/18

2^{x} + 3^{x} = 13

x = \frac{\ln t}{\ln 2}

t^{\frac{\ln 3}{\ln 2}} + t = 13

since \frac{\ln 3}{\ln 2} is irrational we can only approximate

Commented by Tawa1 last updated on 17/Aug/18

God bless you sir

Facebook Tweet Pin