Trigonometry Examples

$\log (x + 11) - \log (2) = \log (5 x + 3)$

Step 1

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

$\log (\frac{x + 11}{2}) = \log (5 x + 3)$

Step 2

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

$\frac{x + 11}{2} = 5 x + 3$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $2$ .

$\frac{x + 11}{2} \cdot 2 = (5 x + 3) \cdot 2$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{x + 11}{2} \cdot 2 = (5 x + 3) \cdot 2$

Step 3.2.1.1.2

Rewrite the expression.

$x + 11 = (5 x + 3) \cdot 2$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(5 x + 3) \cdot 2$ .

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Step 3.2.2.1.1

Apply the distributive property.

$x + 11 = 5 x \cdot 2 + 3 \cdot 2$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $2$ by $5$ .

$x + 11 = 10 x + 3 \cdot 2$

Step 3.2.2.1.2.2

Multiply $3$ by $2$ .

$x + 11 = 10 x + 6$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $10 x$ from both sides of the equation.

$x + 11 - 10 x = 6$

Step 3.3.1.2

Subtract $10 x$ from $x$ .

$- 9 x + 11 = 6$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $11$ from both sides of the equation.

$- 9 x = 6 - 11$

Step 3.3.2.2

Subtract $11$ from $6$ .

$- 9 x = - 5$

Step 3.3.3

Divide each term in $- 9 x = - 5$ by $- 9$ and simplify.

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Step 3.3.3.1

Divide each term in $- 9 x = - 5$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{- 5}{- 9}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 9$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{- 5}{- 9}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 9}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{5}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{9}$

Decimal Form:

$x = 0.\overline{5}$