Trigonometry Examples

$\log_{3} (x + 1) = 2 + \log_{3} (x - 3)$

Step 1

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

$\log_{3} (x + 1) - \log_{3} (x - 3) = 2$

Step 2

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

$\log_{3} (\frac{x + 1}{x - 3}) = 2$

Step 3

Rewrite $\log_{3} (\frac{x + 1}{x - 3}) = 2$ 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 $b^{y} = x$ .

$3^{2} = \frac{x + 1}{x - 3}$

Step 4

Cross multiply to remove the fraction.

$x + 1 = 3^{2} (x - 3)$

Step 5

Simplify $3^{2} (x - 3)$ .

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Step 5.1

Raise $3$ to the power of $2$ .

$x + 1 = 9 (x - 3)$

Step 5.2

Apply the distributive property.

$x + 1 = 9 x + 9 \cdot - 3$

Step 5.3

Multiply $9$ by $- 3$ .

$x + 1 = 9 x - 27$

Step 6

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

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Step 6.1

Subtract $9 x$ from both sides of the equation.

$x + 1 - 9 x = - 27$

Step 6.2

Subtract $9 x$ from $x$ .

$- 8 x + 1 = - 27$

Step 7

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1

Subtract $1$ from both sides of the equation.

$- 8 x = - 27 - 1$

Step 7.2

Subtract $1$ from $- 27$ .

$- 8 x = - 28$

Step 8

Divide each term in $- 8 x = - 28$ by $- 8$ and simplify.

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Step 8.1

Divide each term in $- 8 x = - 28$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 28}{- 8}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $- 8$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 28}{- 8}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 28}{- 8}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $- 28$ and $- 8$ .

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Step 8.3.1.1

Factor $- 4$ out of $- 28$ .

$x = \frac{- 4 (7)}{- 8}$

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Factor $- 4$ out of $- 8$ .

$x = \frac{- 4 \cdot 7}{- 4 \cdot 2}$

Step 8.3.1.2.2

Cancel the common factor.

$x = \frac{- 4 \cdot 7}{- 4 \cdot 2}$

Step 8.3.1.2.3

Rewrite the expression.

$x = \frac{7}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{2}$

Decimal Form:

$x = 3.5$

Mixed Number Form:

$x = 3 \frac{1}{2}$