Precalculus Examples

$\log_{3} (8) - \log_{3} (x + 10) = \log_{3} (72)$

Step 1

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

$\log_{3} (\frac{8}{x + 10}) = \log_{3} (72)$

Step 2

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

$\frac{8}{x + 10} = 72$

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.

$x + 10, 1$

Step 3.1.2

Remove parentheses.

$x + 10, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x + 10$

Step 3.2

Multiply each term in $\frac{8}{x + 10} = 72$ by $x + 10$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{8}{x + 10} = 72$ by $x + 10$ .

$\frac{8}{x + 10} (x + 10) = 72 (x + 10)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x + 10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{8}{x + 10} (x + 10) = 72 (x + 10)$

Step 3.2.2.1.2

Rewrite the expression.

$8 = 72 (x + 10)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Apply the distributive property.

$8 = 72 x + 72 \cdot 10$

Step 3.2.3.2

Multiply $72$ by $10$ .

$8 = 72 x + 720$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $72 x + 720 = 8$ .

$72 x + 720 = 8$

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 $720$ from both sides of the equation.

$72 x = 8 - 720$

Step 3.3.2.2

Subtract $720$ from $8$ .

$72 x = - 712$

Step 3.3.3

Divide each term in $72 x = - 712$ by $72$ and simplify.

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Step 3.3.3.1

Divide each term in $72 x = - 712$ by $72$ .

$\frac{72 x}{72} = \frac{- 712}{72}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $72$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{72 x}{72} = \frac{- 712}{72}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 712}{72}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Cancel the common factor of $- 712$ and $72$ .

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Step 3.3.3.3.1.1

Factor $8$ out of $- 712$ .

$x = \frac{8 (- 89)}{72}$

Step 3.3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.3.1.2.1

Factor $8$ out of $72$ .

$x = \frac{8 \cdot - 89}{8 \cdot 9}$

Step 3.3.3.3.1.2.2

Cancel the common factor.

$x = \frac{8 \cdot - 89}{8 \cdot 9}$

Step 3.3.3.3.1.2.3

Rewrite the expression.

$x = \frac{- 89}{9}$

Step 3.3.3.3.2

Move the negative in front of the fraction.

$x = - \frac{89}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{89}{9}$

Decimal Form:

$x = - 9.\overline{8}$

Mixed Number Form:

$x = - 9 \frac{8}{9}$