Solve 5^x=2x+1 or Prove Impossibility

[Little ant]

Find the solutions of 5^x=2x+1 by ordinary methods? if it can´t be found by these ways, then prove that it's imposible.

 

[stevenb]

It depends on what you mean by ordinary methods.

By inspection, x=0.

 

[Little ant]

but, there is other answer, how you find it?

 

[Femme_physics]

The exponent ^5 is behind the X or after?

 

[JJacquelin]

Ordinary method consists in studying the function y=(5^x)-(2x+1)
This shows that two roots exist (one of them is obvious).
Drawing the graph of the function allows to obtain a first approximate of the second root.
Then, numerical computation leads to the value of the root, as accurate as we want. There are a lot of numerical methods : Newton-Raphson and many other...
Analytical solving is outside the scope of ordinary methods. It requires the use of a special function : the Lambert W function.