Question 417286
You're given an exponential function (which is transcendental) and an algebraic function in the same equation. Transcendental functions do not combine well with algebraic functions. The best way (other than using a calculator) is to use Newton's method.


Newton's method defines a recursive sequence {{{x[i]}}} with an initial "guess" {{{x[0]}}}. Newton's method says that, to find the zeros of a function f(x) = 0 with a given {{{x[0]}}},


{{{x[i+1] = x[i] - (f(x[i])/f^(1) (x[i]))}}}


Here, {{{f^(1) (x[i])}}} represents f'(x[i]), the derivative of the function {{{f(x) = a^x - bx - 1}}}. Here, the derivative of f(x) is {{{f^(1)(x) =(ln(a))a^x - b}}}.