Evaluate log base 3 of 2x^2=2 log base 3 of 2x

$\log_{3} (2 x^{2}) = 2 \log_{3} (2 x)$

Step 1

Simplify $2 \log_{3} (2 x)$ by moving $2$ inside the logarithm.

$\log_{3} (2 x^{2}) = \log_{3} ({(2 x)}^{2})$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$2 x^{2} = {(2 x)}^{2}$

Step 3

Solve for $x$ .

Simplify ${(2 x)}^{2}$ .

Apply the product rule to $2 x$ .

$2 x^{2} = 2^{2} x^{2}$

Raise $2$ to the power of $2$ .

$2 x^{2} = 4 x^{2}$

Move all terms containing $x$ to the left side of the equation.

Subtract $4 x^{2}$ from both sides of the equation.

$2 x^{2} - 4 x^{2} = 0$

Subtract $4 x^{2}$ from $2 x^{2}$ .

$- 2 x^{2} = 0$

Divide each term in $- 2 x^{2} = 0$ by $- 2$ and simplify.

Divide each term in $- 2 x^{2} = 0$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{0}{- 2}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

$x^{2} = 0$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Exclude the solutions that do not make $\log_{3} (2 x^{2}) = 2 \log_{3} (2 x)$ true.

$No solution$