a^x = log _a (x) a=?


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Question Number 92135 by jagoll last updated on 05/May/20

$a^{x} = \log_{a} (x)$ $a = ?$
Commented by Tony Lin last updated on 05/May/20
$when 0<x<e^(−e) , x has three solutions when e^(−e) ≤a<1, x has one solution when 1<a< e^(1/e) , x has two solutions when a=e^(1/e) , x has one solution when a>e^(1/e) , x has no solutions ∴ when a^x =log_a x ⇒aε(0,e^(1/e) ]\{1}$
$w h e n 0 < x < e^{- e}, x h a s t h r e e s o l u t i o n s$ $w h e n e^{- e} ⩽ a < 1, x h a s o n e s o l u t i o n$ $w h e n 1 < a < e^{\frac{1}{e}}, x h a s t w o s o l u t i o n s$ $w h e n a = e^{\frac{1}{e}}, x h a s o n e s o l u t i o n$ $w h e n a > e^{\frac{1}{e}}, x h a s n o s o l u t i o n s$ $∴ w h e n a^{x} = {l o g}_{a} x$ $\Rightarrow a ϵ (0, e^{\frac{1}{e}}] ∖ {1}$
Commented by mr W last updated on 05/May/20

$a = 1 \Rightarrow o n e s o l u t i o n x = 1$
Commented by jagoll last updated on 05/May/20

$\log_{1} (1) defined ?$
Commented by MJS last updated on 05/May/20

$\lim_{x \to 1} \log_{x} x = \lim_{x \to 1} \frac{\ln x}{\ln x} = \lim_{x \to 1} 1 = 1$ $also \log_{0} 0 = 1$ $\log_{z} z = 1 \forall z \in C$
Commented by Tony Lin last updated on 05/May/20

$a^{a^{x}} = x$ $a^{x} l n a = l n x$ $(x l n a) e^{x l n a} = x l n x$ $x l n a = W (x l n x)$ $a = e^{\frac{W (x l n x)}{x}}$
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