Algebra Examples

$\ln (x) - \ln (x + 2) = \ln (8 x)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 2

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

$\frac{x}{x + 2} = 8 x$

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + 2, 1$

Step 3.1.2

Remove parentheses.

$x + 2, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x + 2$

Step 3.2

Multiply each term in $\frac{x}{x + 2} = 8 x$ by $x + 2$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{x}{x + 2} = 8 x$ by $x + 2$ .

$\frac{x}{x + 2} (x + 2) = 8 x (x + 2)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x + 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{x}{x + 2} (x + 2) = 8 x (x + 2)$

Step 3.2.2.1.2

Rewrite the expression.

$x = 8 x (x + 2)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Apply the distributive property.

$x = 8 x \cdot x + 8 x \cdot 2$

Step 3.2.3.2

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

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Step 3.2.3.2.1

Move $x$ .

$x = 8 (x \cdot x) + 8 x \cdot 2$

Step 3.2.3.2.2

Multiply $x$ by $x$ .

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

Step 3.2.3.3

Multiply $2$ by $8$ .

$x = 8 x^{2} + 16 x$

Step 3.3

Solve the equation.

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Step 3.3.1

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

$8 x^{2} + 16 x = x$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $x$ from both sides of the equation.

$8 x^{2} + 16 x - x = 0$

Step 3.3.2.2

Subtract $x$ from $16 x$ .

$8 x^{2} + 15 x = 0$

Step 3.3.3

Factor $x$ out of $8 x^{2} + 15 x$ .

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Step 3.3.3.1

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

$x (8 x) + 15 x = 0$

Step 3.3.3.2

Factor $x$ out of $15 x$ .

$x (8 x) + x \cdot 15 = 0$

Step 3.3.3.3

Factor $x$ out of $x (8 x) + x \cdot 15$ .

$x (8 x + 15) = 0$

Step 3.3.4

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

$x = 0$

$8 x + 15 = 0$

Step 3.3.5

Set $x$ equal to $0$ .

$x = 0$

Step 3.3.6

Set $8 x + 15$ equal to $0$ and solve for $x$ .

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Step 3.3.6.1

Set $8 x + 15$ equal to $0$ .

$8 x + 15 = 0$

Step 3.3.6.2

Solve $8 x + 15 = 0$ for $x$ .

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Step 3.3.6.2.1

Subtract $15$ from both sides of the equation.

$8 x = - 15$

Step 3.3.6.2.2

Divide each term in $8 x = - 15$ by $8$ and simplify.

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Step 3.3.6.2.2.1

Divide each term in $8 x = - 15$ by $8$ .

$\frac{8 x}{8} = \frac{- 15}{8}$

Step 3.3.6.2.2.2

Simplify the left side.

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Step 3.3.6.2.2.2.1

Cancel the common factor of $8$ .

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Step 3.3.6.2.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 15}{8}$

Step 3.3.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{8}$

Step 3.3.6.2.2.3

Simplify the right side.

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Step 3.3.6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{15}{8}$

Step 3.3.7

The final solution is all the values that make $x (8 x + 15) = 0$ true.

$x = 0, - \frac{15}{8}$

Step 4

Exclude the solutions that do not make $\ln (x) - \ln (x + 2) = \ln (8 x)$ true.

$No solution$