Algebra Examples

$2 \log (x) - 2 \log (x + 1) = 0$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \log (x) - 2 \log (x + 1) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log (x) - 2 \log (x + 1)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$\log (x^{2}) - 2 \log (x + 1) = 0$

Step 2.1.1.2

Simplify $- 2 \log (x + 1)$ by moving $2$ inside the logarithm.

$\log (x^{2}) - \log ({(x + 1)}^{2}) = 0$

Step 2.1.2

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

$\log (\frac{x^{2}}{{(x + 1)}^{2}}) = 0$

Step 3

Rewrite $\log (\frac{x^{2}}{{(x + 1)}^{2}}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{0} = \frac{x^{2}}{{(x + 1)}^{2}}$

Step 4

Cross multiply to remove the fraction.

$x^{2} = 10^{0} ({(x + 1)}^{2})$

Step 5

Simplify $10^{0} ({(x + 1)}^{2})$ .

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Step 5.1

Simplify the expression.

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Step 5.1.1

Anything raised to $0$ is $1$ .

$x^{2} = 1 {(x + 1)}^{2}$

Step 5.1.2

Multiply ${(x + 1)}^{2}$ by $1$ .

$x^{2} = {(x + 1)}^{2}$

Step 5.1.3

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$x^{2} = (x + 1) (x + 1)$

Step 5.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$x^{2} = x (x + 1) + 1 (x + 1)$

Step 5.2.2

Apply the distributive property.

$x^{2} = x \cdot x + x \cdot 1 + 1 (x + 1)$

Step 5.2.3

Apply the distributive property.

$x^{2} = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $x$ by $x$ .

$x^{2} = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Step 5.3.1.2

Multiply $x$ by $1$ .

$x^{2} = x^{2} + x + 1 x + 1 \cdot 1$

Step 5.3.1.3

Multiply $x$ by $1$ .

$x^{2} = x^{2} + x + x + 1 \cdot 1$

Step 5.3.1.4

Multiply $1$ by $1$ .

$x^{2} = x^{2} + x + x + 1$

Step 5.3.2

Add $x$ and $x$ .

$x^{2} = x^{2} + 2 x + 1$

Step 6

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

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Step 6.1

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

$x^{2} - x^{2} = 2 x + 1$

Step 6.2

Subtract $2 x$ from both sides of the equation.

$x^{2} - x^{2} - 2 x = 1$

Step 6.3

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

$0 - 2 x = 1$

Step 6.4

Subtract $2 x$ from $0$ .

$- 2 x = 1$

Step 7

Divide each term in $- 2 x = 1$ by $- 2$ and simplify.

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Step 7.1

Divide each term in $- 2 x = 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{1}{- 2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{1}{- 2}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{- 2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 8

Exclude the solutions that do not make $2 \log (x) - 2 \log (x + 1) = 0$ true.

$No solution$