Trigonometry Examples

$\ln (5 x) - \ln (x + 2) = \ln (8)$

Step 1

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

$\ln (\frac{5 x}{x + 2}) = \ln (8)$

Step 2

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

$\frac{5 x}{x + 2} = 8$

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + 2, 1$

Step 3.1.2

Remove parentheses.

$x + 2, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x + 2$

Step 3.2

Multiply each term in $\frac{5 x}{x + 2} = 8$ by $x + 2$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{5 x}{x + 2} = 8$ by $x + 2$ .

$\frac{5 x}{x + 2} (x + 2) = 8 (x + 2)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x + 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{x + 2} (x + 2) = 8 (x + 2)$

Step 3.2.2.1.2

Rewrite the expression.

$5 x = 8 (x + 2)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Apply the distributive property.

$5 x = 8 x + 8 \cdot 2$

Step 3.2.3.2

Multiply $8$ by $2$ .

$5 x = 8 x + 16$

Step 3.3

Solve the equation.

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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.

$5 x - 8 x = 16$

Step 3.3.1.2

Subtract $8 x$ from $5 x$ .

$- 3 x = 16$

Step 3.3.2

Divide each term in $- 3 x = 16$ by $- 3$ and simplify.

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Step 3.3.2.1

Divide each term in $- 3 x = 16$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{16}{- 3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{16}{- 3}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{- 3}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{16}{3}$

Step 4

Exclude the solutions that do not make $\ln (5 x) - \ln (x + 2) = \ln (8)$ true.

$No solution$