Finite Math Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 2

Simplify the right side.

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Step 2.1

Simplify $\ln (x + 4) + \ln (2 x)$ .

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Step 2.1.1

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

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

Step 2.1.2

Apply the distributive property.

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

Step 2.1.3

Simplify the expression.

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Step 2.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.3.2

Multiply $2$ by $4$ .

$\ln (x^{2}) = \ln (2 x \cdot x + 8 x)$

Step 2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.1

Move $x$ .

$\ln (x^{2}) = \ln (2 (x \cdot x) + 8 x)$

Step 2.1.4.2

Multiply $x$ by $x$ .

$\ln (x^{2}) = \ln (2 x^{2} + 8 x)$

Step 3

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

$x^{2} = 2 x^{2} + 8 x$

Step 4

Solve for $x$ .

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Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} + 8 x = x^{2}$

Step 4.2

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

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Step 4.2.1

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

$2 x^{2} + 8 x - x^{2} = 0$

Step 4.2.2

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

$x^{2} + 8 x = 0$

Step 4.3

Factor $x$ out of $x^{2} + 8 x$ .

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Step 4.3.1

Factor $x$ out of $x^{2}$ .

$x \cdot x + 8 x = 0$

Step 4.3.2

Factor $x$ out of $8 x$ .

$x \cdot x + x \cdot 8 = 0$

Step 4.3.3

Factor $x$ out of $x \cdot x + x \cdot 8$ .

$x (x + 8) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 8 = 0$

Step 4.5

Set $x$ equal to $0$ .

$x = 0$

Step 4.6

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 4.6.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 4.7

The final solution is all the values that make $x (x + 8) = 0$ true.

$x = 0, - 8$

Step 5

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

$No solution$