Question

In the following exercises, solve each equation.

$3^{x^2-2x}=\frac{1}{3}$

Short answer

The answer is$x=1$

Step-by-step solution

Step 1. Given

The equation is$3^{x^2-2x}=\frac{1}{3}$

Step 2. Solve the equation

$$\begin{gathered} 3^{x^2-2x}=\frac{1}{3} \\ {3}^{x^2-2x}=3^{-1} \end{gathered}$$

Bases are same so the exponents should be equal.

$$\begin{gathered} x^2-2x=-1 \\ {x}^2-2x+1=0 \\ {\left(x-1\right)}^2=0 \\ x-1=0 \\ x=1 \end{gathered}$$

Step 3. Check the answer

$$\begin{gathered} 3^{x^2-2x}=\frac{1}{3} \\ {3}^{{\left(1\right)}^2-2\left(1\right)}=\frac{1}{3} \\ {3}^{1-2}=\frac{1}{3} \\ {3}^{-1}=\frac{1}{3} \\ \frac{1}{3}=\frac{1}{3} \end{gathered}$$

True