Question

Solve for x: x^4 = 3^x

Answer 1

There are no algebraic methods for finding solutions to a general mix of exponential and polynomial terms. A graphing calculator can be helpful.

This equation has 3 real solutions, approximately ...
  x ∈ {-0.802246431546, 1.51677641228, 7.17475582739}

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In the folder "iteration for solutions" is an equation for Newton's method iteration, essentially, ...
  g(x) = x -f(x)/f'(x)
where f(x) is defined as shown in the picture.

Many graphing calculators can compute a numerical derivative, so you can essentially write the formula in this form without having to do the derivative-taking yourself. This calculator is nicely interactive, so the iteration result is produced at the same time the argument for g(x) is entered. Essentially, you write the answer by copying the answer using the 4-digit zero-crossing values shown on the graph as the iteration starting point.