Intersection of y=log_2(x), y=ax

Question

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.

awang66 · Accepted Answer

Dear Hamish Arblaster, here is a solution:

Hamish Arblaster · Answer

The answer is this comment, but I can't mark comments as answers. https://www.blackpenredpen.com/discussions/main/comment/5ea70112bdd4a40017453622