Question

Solve the valid solutions of the following equations providing your values for some of them. Write your answers in solution set form. Identify the extraneous solution, if there is.

a^x-1 = b^3-x

(provide values for a and b which are different integers greater than 1)

Answer 1

$a^{x-1}=b^{3-x}$

$\ln a*(x-1)=\ln b* (3-x)$

$x=\frac{3\ln a+\ln b}{\ln a+\\ln b}$

For a=2, b=4:

$x=\frac{3\ln 2+\ln 4}{\ln 2+\\ln 4}=\frac{5\ln 2}{3\ln 2}=\frac{5}{3}$