Hamish Arblaster
Apr 27
Edited: Apr 27

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

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.

1
awang66
Apr 27

Dear Hamish Arblaster, here is a solution:



Hamish Arblaster
Apr 27Edited: Apr 27

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

Sorry for the confusion.

Ian Fowler
Apr 27Edited: Apr 27

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:


1
Ian Fowler
Apr 27Edited: Apr 27

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?