Precalculus Examples

$\log (6 x + 9) = 1 + \log (x - 1)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log (6 x + 9) - \log (x - 1) = 1$

Step 2

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

$\log (\frac{6 x + 9}{x - 1}) = 1$

Step 3

Factor $3$ out of $6 x + 9$ .

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Step 3.1

Factor $3$ out of $6 x$ .

$\log (\frac{3 (2 x) + 9}{x - 1}) = 1$

Step 3.2

Factor $3$ out of $9$ .

$\log (\frac{3 (2 x) + 3 (3)}{x - 1}) = 1$

Step 3.3

Factor $3$ out of $3 (2 x) + 3 (3)$ .

$\log (\frac{3 (2 x + 3)}{x - 1}) = 1$

Step 4

Rewrite $\log (\frac{3 (2 x + 3)}{x - 1}) = 1$ 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 $b^{y} = x$ .

$10^{1} = \frac{3 (2 x + 3)}{x - 1}$

Step 5

Cross multiply to remove the fraction.

$3 (2 x + 3) = 10 (x - 1)$

Step 6

Simplify $10 (x - 1)$ .

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Step 6.1

Apply the distributive property.

$3 (2 x + 3) = 10 x + 10 \cdot - 1$

Step 6.2

Multiply $10$ by $- 1$ .

$3 (2 x + 3) = 10 x - 10$

Step 7

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

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Step 7.1

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

$3 (2 x + 3) - 10 x = - 10$

Step 7.2

Simplify each term.

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Step 7.2.1

Apply the distributive property.

$3 (2 x) + 3 \cdot 3 - 10 x = - 10$

Step 7.2.2

Multiply $2$ by $3$ .

$6 x + 3 \cdot 3 - 10 x = - 10$

Step 7.2.3

Multiply $3$ by $3$ .

$6 x + 9 - 10 x = - 10$

Step 7.3

Subtract $10 x$ from $6 x$ .

$- 4 x + 9 = - 10$

Step 8

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

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Step 8.1

Subtract $9$ from both sides of the equation.

$- 4 x = - 10 - 9$

Step 8.2

Subtract $9$ from $- 10$ .

$- 4 x = - 19$

Step 9

Divide each term in $- 4 x = - 19$ by $- 4$ and simplify.

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Step 9.1

Divide each term in $- 4 x = - 19$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 19}{- 4}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 4$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 19}{- 4}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 19}{- 4}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Dividing two negative values results in a positive value.

$x = \frac{19}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19}{4}$

Decimal Form:

$x = 4.75$

Mixed Number Form:

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