Find All Complex Solutions 2 natural log of x=4 natural log of 2

$2 \ln (x) = 4 \ln (2)$

Step 1

Simplify.

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$\ln (x^{2}) = 4 \ln (2)$

Simplify $4 \ln (2)$ by moving $4$ inside the logarithm.

$\ln (x^{2}) = \ln (2^{4})$

Step 2

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

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

Step 3

Solve for $x$ .

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

$x = \pm \sqrt{2^{4}}$

Simplify $\pm \sqrt{2^{4}}$ .

Raise $2$ to the power of $4$ .

$x = \pm \sqrt{16}$

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

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

$x = \pm 4$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$

Step 4

Exclude the solutions that do not make $2 \ln (x) = 4 \ln (2)$ true.

$x = 4$