, .
Knowing there will only be one root from the properties of exponential functions, we could use a root finding technique, such as Newton's method, which converges to in 10 iterations on my calculator with an initial guess of .
I would use inspection first to see if a rational root exists, but say the equation had instead been:
Now, using Newton's method, we find:
Have you studied differential calculus? Newton's method uses differentiation in its recursive approach. Newton's method essentially finds the x-intercept of the line tangent to the function for which we are trying to find a root at a value of x and uses that as a better approximation. Then this method is repeated recursively to converge to a root of the function. There are functions for which this doesn't always work for particular values of x, but for well-behaved functions like exponential functions, this is a good technique to use.
You have the correct derivative.
For the function:
and so by Newton's method, we have the recursion:
As stated, when there certainly is no need for this technique, but I simply wanted to give a method for approximating the root when it is irrational, since you seemed to be interested in an "algebraic" method.