Trigonometry Examples

$\log_{6} (x + 7) - \log_{6} (4) = \log_{6} (2 x + 1)$

Step 1

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

$\log_{6} (\frac{x + 7}{4}) = \log_{6} (2 x + 1)$

Step 2

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

$\frac{x + 7}{4} = 2 x + 1$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $4$ .

$\frac{x + 7}{4} \cdot 4 = (2 x + 1) \cdot 4$

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 $4$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{x + 7}{4} \cdot 4 = (2 x + 1) \cdot 4$

Step 3.2.1.1.2

Rewrite the expression.

$x + 7 = (2 x + 1) \cdot 4$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(2 x + 1) \cdot 4$ .

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Step 3.2.2.1.1

Apply the distributive property.

$x + 7 = 2 x \cdot 4 + 1 \cdot 4$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $4$ by $2$ .

$x + 7 = 8 x + 1 \cdot 4$

Step 3.2.2.1.2.2

Multiply $4$ by $1$ .

$x + 7 = 8 x + 4$

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 $8 x$ from both sides of the equation.

$x + 7 - 8 x = 4$

Step 3.3.1.2

Subtract $8 x$ from $x$ .

$- 7 x + 7 = 4$

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 $7$ from both sides of the equation.

$- 7 x = 4 - 7$

Step 3.3.2.2

Subtract $7$ from $4$ .

$- 7 x = - 3$

Step 3.3.3

Divide each term in $- 7 x = - 3$ by $- 7$ and simplify.

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Step 3.3.3.1

Divide each term in $- 7 x = - 3$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 3}{- 7}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 7$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 3}{- 7}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 7}$

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.

$x = \frac{3}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{7}$

Decimal Form:

$x = 0.\overline{428571}$