I have 2 equations: *y=log_2(x)*, *y=ax*

I would like to solve for a where they intersect at exactly one point over the reals.

Graphically I can determine that *a* must be somewhere between 0.53072 and 0.53077.

I do not know how to go about solving an equation like this, please help me.

I would like the working / a method of solving which gives the answer in exact form.

Usage of advanced functions such as the Lambert W function is perfectly fine.

Thanks!

Edit (I also changed it to *a*, instead of *x*, which is what I meant):

Graph:

I want *a* to cause log_2(x), y=ax to touch at exactly 1 point.

The answer is this comment, but I can't mark comments as answers.

https://www.blackpenredpen.com/discussions/main/comment/5ea70112bdd4a40017453622This raises an interesting question: If a > 1/eln(2) then the line will not intersect the log curve at all. How does this reconcile with the x-value arrived at in the above? I did it the same way using ln(1/x) and got the same answer. Good solution awang66. Are there some domain restrictions on W?

Dear Hamish Arblaster, here is a solution: