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/5ea70112bdd4a40017453622
Dear Hamish Arblaster, here is a solution:
Thank you. I am sorry, I was meant to ask to solve for a, can you do that too please?
Sorry for the confusion.
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:
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?