Equation with the same base

Problem

Solve the exponential equation $3^{2x-1} = 27^{x-2}.$

Solution

Since $27 = 3^3$, we have $27^{x-2} = 3^{3(x-2)} = 3^{3x-6}$. By injectivity of the exponential function the exponents are set equal: $2x - 1 = 3x - 6 \iff -x = -5 \iff \boxed{x = 5}.$

Links

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