Techniques for Solving Exponential Equations

Example

$2^x+2^{x+1}=9$

Solution - Factor

$2^x(1+2)=9$ $2^x(3)=9$ $2^x=3$ $x=\frac{{\ln }_3}{{\ln }_2}$

Example 2

$3^x-3^{-x}=2$ let $3^x=A$ $A-\frac{1}{A}=2$ $A^2-1=2A$ $A^2-2A-1$ $A=1\pm \sqrt{2}$ $x=\frac{\ln (1+\sqrt{2}}{{\ln }_3}$

Example 3

$4^{3x-1}=3^{x+4}$

Log Both Sides

$\log (4^{3x-1})=\log (3^{x+4})$ $(3x-1)\log (4)=(x+4)\log (3)$ $3x\log (4)-\log (4)=x\log (3)+4\log (3)$ $3x\log (4)-x\log (3)=4\log (3)+\log (4)$

Common Base

$3^{1.261859507(3x-1)}=3^{x+4}$ $3.785578521x+1.261859507=x+4$ $2.785578521x=2.738140493$ $x=0.982970134$