a-x-log-a-x-a-

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

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}}