Exponential equation by substitution

Text

Solve the equation $4^{x}-5\cdot 2^{x}+4=0$.

Solution

Note that $4^{x}=(2^{x})^{2}$. Set $t=2^{x}$, with $t>0$; the equation becomes $t^{2}-5t+4=0\;\Rightarrow\;(t-1)(t-4)=0\;\Rightarrow\; t=1\ \lor\ t=4.$ Return to $x$:

• $2^{x}=1\Rightarrow x=0$;
• $2^{x}=4=2^{2}\Rightarrow x=2$. $\boxed{\,x=0\quad\lor\quad x=2\,}$