Calculus Examples

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

Step 1

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

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

Step 2

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

$\frac{3 + x}{x - 5} = 3$

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.

$x - 5, 1$

Step 3.1.2

Remove parentheses.

$x - 5, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x - 5$

Step 3.2

Multiply each term in $\frac{3 + x}{x - 5} = 3$ by $x - 5$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{3 + x}{x - 5} = 3$ by $x - 5$ .

$\frac{3 + x}{x - 5} (x - 5) = 3 (x - 5)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x - 5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 + x}{x - 5} (x - 5) = 3 (x - 5)$

Step 3.2.2.1.2

Rewrite the expression.

$3 + x = 3 (x - 5)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Apply the distributive property.

$3 + x = 3 x + 3 \cdot - 5$

Step 3.2.3.2

Multiply $3$ by $- 5$ .

$3 + x = 3 x - 15$

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 $3 x$ from both sides of the equation.

$3 + x - 3 x = - 15$

Step 3.3.1.2

Subtract $3 x$ from $x$ .

$3 - 2 x = - 15$

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 $3$ from both sides of the equation.

$- 2 x = - 15 - 3$

Step 3.3.2.2

Subtract $3$ from $- 15$ .

$- 2 x = - 18$

Step 3.3.3

Divide each term in $- 2 x = - 18$ by $- 2$ and simplify.

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Step 3.3.3.1

Divide each term in $- 2 x = - 18$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 18}{- 2}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 2$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 18}{- 2}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 18}{- 2}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 18$ by $- 2$ .

$x = 9$