Circular logic with exponentials

[Oaky]

Just wondering how you would go about solving an equation something like...

\(\displaystyle 3x = e^{2x}\)

which is equivalent to

\(\displaystyle \ln{3x} = 2x\)

and seeing as they're sort of defined in terms of each other, I'm going nowhere.

Help would much be appreciated
Thanks!

 

[Deleted member 4993]

Just wondering how you would go about solving an equation something like...

\(\displaystyle 3x = e^{2x}\)

which is equivalent to

\(\displaystyle \ln{3x} = 2x\)

and seeing as they're sort of defined in terms of each other, I'm going nowhere.

Help would much be appreciated
Thanks!

You could solve it graphically - find point/s of intersetion of y = 3x and y = e^2x

Also using numerical methods, you could find roots of f(x) = e^2x - 3x

 

[Oaky]

I can get approximate (decimal) answers from my calculator, but what algebraic methods would I use to solve it by hand?
You say to find the roots, but I'm unsure of how to do this manually.

 

[Deleted member 4993]

I can get approximate (decimal) answers from my calculator, but what algebraic methods would I use to solve it by hand?
You say to find the roots, but I'm unsure of how to do this manually.

As far as I can see, there is no closed form solution - hence I do not see any "simple" solution.