Roots of an Exponential Equation


Consider the equation --

If a plot is made of a function determined by each side of the equation then the roots of the given equation will be determined by where the two curves intersect. From the graph at the right one can see there are roots at approximately

x = -0.94 and x = 1.08

Are these the only roots?





The graph of

shows similar information.   Again, the approximate roots can be inferred from where the graph crosses the x-axis.  These graphs can be readily produced with a hand-held calculator such as the TI-83 or any graphing software.   

But, do they convey all of the roots?




Consider taking the logarithm to some base of each side of the equation and make graphs of the log functions.



Prove:  There are 3 roots to the equation


Find approximations to the three roots using iteration (perhaps implemented on a spreadsheet).