Elementary exponential equation

Text

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

Solution

Write the right-hand side as a power of $3$: $27=3^{3}$. The equation becomes $3^{\,2x-1}=3^{3}.$ The exponential is injective, so we may equate the exponents: $2x-1=3\;\Rightarrow\; 2x=4\;\Rightarrow\; x=2.$ $\boxed{\,x=2\,}$