Algebra Examples

$\log (x + 12) = \log (x) + \log (12)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log (x) + \log (12)$ .

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Step 1.1.1

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

$\log (x + 12) = \log (x \cdot 12)$

Step 1.1.2

Move $12$ to the left of $x$ .

$\log (x + 12) = \log (12 x)$

Step 2

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

$x + 12 = 12 x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $12 x$ from both sides of the equation.

$x + 12 - 12 x = 0$

Step 3.1.2

Subtract $12 x$ from $x$ .

$- 11 x + 12 = 0$

Step 3.2

Subtract $12$ from both sides of the equation.

$- 11 x = - 12$

Step 3.3

Divide each term in $- 11 x = - 12$ by $- 11$ and simplify.

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Step 3.3.1

Divide each term in $- 11 x = - 12$ by $- 11$ .

$\frac{- 11 x}{- 11} = \frac{- 12}{- 11}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 11$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 11 x}{- 11} = \frac{- 12}{- 11}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 12}{- 11}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{12}{11}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{12}{11}$

Decimal Form:

$x = 1.\overline{09}$

Mixed Number Form:

$x = 1 \frac{1}{11}$