Determine all positive numbers a for which the curve y=a:x intersects the line y=x without calculus The answer is 0<a<e:(1/e) , but how to find it? Is it a system of equations? Which ones? I just need an idea at least, because I'm stuck. If it is impossible without calculus, solve it with it.

aanpalendmw

aanpalendmw

Answered question

2022-07-19

Determine all positive numbers a for which the curve y = a x intersects the line y = x without calculus
The answer is 0 < a < e 1 / e , but how to find it? Is it a system of equations? Which ones? I just need an idea at least, because I'm stuck. If it is impossible without calculus, solve it with it.

Answer & Explanation

Osvaldo Crosby

Osvaldo Crosby

Beginner2022-07-20Added 12 answers

We have
a x = x x ln ( a ) = ln ( x ) ln ( x ) x = ln ( a )
This equation is indeed unsolvable without some amount of calculus.
Using calculus, however, we may show that ln ( x ) / x is a function that increases from at x = 0, reaches its maximum of 1 e at x = e, and then decreases as x towards 0
With that in mind, it is clear that the curve y = ln ( a ) will intersect the curve y = ln ( x ) / x if and only if < ln ( a ) 1 e , which is to say that 0 < a e 1 / e
prkosnognm

prkosnognm

Beginner2022-07-21Added 5 answers

This is just a complement to previous answers.
As already told by Omnomnomnom
a x = x x ln ( a ) = ln ( x ) ln ( x ) x = ln ( a )
connot be solved in terms of elementary functions. The only explicit solution is given using Lambert function
x = W ( log ( a ) ) log ( a )
which has been worked by Euler too; in particular, Euler showed that this equation has a vertical asymptote corresponding to a = e 1 e

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