Trigonometry Examples

$\log (6 x + 5) - \log (x - 1) = 1$

Step 1

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

$\log (\frac{6 x + 5}{x - 1}) = 1$

Step 2

Rewrite $\log (\frac{6 x + 5}{x - 1}) = 1$ 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 $b^{y} = x$ .

$10^{1} = \frac{6 x + 5}{x - 1}$

Step 3

Cross multiply to remove the fraction.

$6 x + 5 = 10 (x - 1)$

Step 4

Simplify $10 (x - 1)$ .

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Step 4.1

Apply the distributive property.

$6 x + 5 = 10 x + 10 \cdot - 1$

Step 4.2

Multiply $10$ by $- 1$ .

$6 x + 5 = 10 x - 10$

Step 5

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

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Step 5.1

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

$6 x + 5 - 10 x = - 10$

Step 5.2

Subtract $10 x$ from $6 x$ .

$- 4 x + 5 = - 10$

Step 6

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

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Step 6.1

Subtract $5$ from both sides of the equation.

$- 4 x = - 10 - 5$

Step 6.2

Subtract $5$ from $- 10$ .

$- 4 x = - 15$

Step 7

Divide each term in $- 4 x = - 15$ by $- 4$ and simplify.

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Step 7.1

Divide each term in $- 4 x = - 15$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 15}{- 4}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 4$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 15}{- 4}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{- 4}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Dividing two negative values results in a positive value.

$x = \frac{15}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15}{4}$

Decimal Form:

$x = 3.75$

Mixed Number Form:

$x = 3 \frac{3}{4}$