Solve for x natural log of 20+ natural log of 5=2 natural log of x

$\ln (20) + \ln (5) = 2 \ln (x)$

Step 1

Rewrite the equation as $2 \ln (x) = \ln (20) + \ln (5)$ .

$2 \ln (x) = \ln (20) + \ln (5)$

Step 2

Simplify the left side.

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

$\ln (x^{2}) = \ln (20) + \ln (5)$

Step 3

Simplify the right side.

Simplify $\ln (20) + \ln (5)$ .

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (x^{2}) = \ln (20 \cdot 5)$

Multiply $20$ by $5$ .

$\ln (x^{2}) = \ln (100)$

Step 4

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

$x^{2} = 100$

Step 5

Solve for $x$ .

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

$x = \pm \sqrt{100}$

Simplify $\pm \sqrt{100}$ .

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2}}$

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

$x = \pm 10$

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.

$x = 10$

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

$x = - 10$

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

$x = 10, - 10$

Step 6

Exclude the solutions that do not make $\ln (20) + \ln (5) = 2 \ln (x)$ true.

$x = 10$