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.

Hamish Arblaster · Accepted Answer

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

Ian Fowler · Answer

This 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?

awang66 · Answer

Dear Hamish Arblaster, here is a solution:

Ian Fowler · Answer

I could have just said that y = ax has to be tangent to the log curve - oh well - hindsight.Another interesting tidbit. The intersection point is at (e,1/ln(2)).Try this:

Hamish Arblaster · Answer

Thank you. I am sorry, I was meant to ask to solve for a, can you do that too please?Sorry for the confusion.