How to solve this equation?

Question

2^x+3^x=13

I know that the answer is x=2 but I don't know how to work this problem out. Can someone please help me?

Answer 1

If you have had any calculus, you can solve using Newton's Iterative method

x = x_o - f(x_o)/f'(x_o)

In this problem, x_o is the intial guess, f(x) = 2x + 3x - 13

and f'(x) = ln(2)*2x + ln(3)*3x

for a first guess of xo = 1 x = 2.7086

for a next guess of x_o = 2.7086 x = 2.204

for a next guess of x_o = 2.204 x = 2.012

for a next guess of x_o = 2.012 x = 2.00007

for a next guess of x_o = 2.00007 x = 2