Logarithmic equation by substitution

Statement

Solve $(\log_3 x)^2 - 3\log_3 x + 2 = 0$ using the substitution $t = \log_3 x$.

Solution

EC: $x > 0$.

Substitution $t = \log_3 x$: $t^2 - 3t + 2 = 0 \iff t = 1 \vee t = 2.$

Returning to $x$:

• $\log_3 x = 1 \iff x = 3$;
• $\log_3 x = 2 \iff x = 9$.

Both satisfy the EC $x>0$. Solutions: $\boxed{x = 3 \vee x = 9}$.

Links

Topics: Logarithmic function
Concepts: Existence conditions · Logarithmic equations · Logarithm
Functions: Logarithmic function
Methods: Logarithm equations
Skills: Solving equations
Exercise type: Solving an equation