Solve (1/3)^x = log<a>x

[CP2016]

1. I hit this logarithm mathematics problem.


2. (1/3)^x = log<a>x
For clarifications, <a> means the base a



3. I have used GeoGeBra to graph them and managed to find the intersection.

But is there a solution for the exact value(s) of x?

 

[member 587159]

I think this depends on the value of a.

 

[CP2016]

That is also what I have thought. I think the question is wrongly set because I was using the slider in GeoGebra without which I got no where to go.

But I when tried putting a=3, just an arbitrary one, I could not figure out.

 

[Ray Vickson]

That is also what I have thought. I think the question is wrongly set because I was using the slider in GeoGebra without which I got no where to go.

But I when tried putting a=3, just an arbitrary one, I could not figure out.

The solution involves a non-elementary function, in this case the so-called Lambert W function; see, eg.,
https://en.wikipedia.org/wiki/Lambert_W_function.