If-y-ln-x-and-y-a-x-have-a-solution-then-find-the-range-of-a-1-0-1-2-1-e-e-3-1-e-4-0-1-

Question Number 15184 by Tinkutara last updated on 08/Jun/17

If y = \ln x and y = a^{x} have a solution

then find the range of a .

(1) (0, 1)

(2) (\frac{1}{e}, e)

(3) (1, e)

(4) (0, 1]

Commented by mrW1 last updated on 08/Jun/17

I think none of the answers is right .

example 1 :

a = 1.1 > 1

\ln x = 1 . 1^{x}

we have solution x = 5.02 and x = 6.93

\Rightarrow answer (1) and (4) are wrong .

example 2 :

a = 2 < e

\ln x = 2^{x}

it has no solution

\Rightarrow answer (2) and (3) are wrong .

Commented by Tinkutara last updated on 08/Jun/17

But answer is (4) . When seen from

graphs, they actually intersect . But

how to find the solution algebrically,

I {don}^{'} t know .

Commented by mrW1 last updated on 08/Jun/17

the \max . value of a is not to calculate

algebrically . it is about 1.10215 . so

the range of a should be

(0, 1.10215]

Commented by Tinkutara last updated on 08/Jun/17

Thanks Sir!