Equation by substitution

Problem

Solve the equation $9^x - 4\cdot 3^x + 3 = 0.$

Solution

Observe that $9^x = (3^2)^x = (3^x)^2$. Setting $t = 3^x$ with $t > 0$: $t^2 - 4t + 3 = 0 \iff (t-1)(t-3) = 0 \iff t = 1 \ \vee\ t = 3.$ Both roots are positive, hence acceptable. Returning to $x$:

• $3^x = 1 = 3^0 \implies x = 0$;
• $3^x = 3 = 3^1 \implies x = 1$.

Solutions: $\boxed{x = 0 \ \vee\ x = 1}$.

Links

Topics: Exponential function
Functions: Exponential function
Methods: Exponential equations
Skills: Solving equations
Exercise type: Solving an equation