Algebra Examples

$\log_{3} (x + 4) - \log_{3} (x + 2) = \log_{3} (27)$

Step 1

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

$\log_{3} (\frac{x + 4}{x + 2}) = \log_{3} (27)$

Step 2

Logarithm base $3$ of $27$ is $3$ .

Tap for more steps...

Step 2.1

Rewrite as an equation.

$\log_{3} (27) = x$

Step 2.2

Rewrite $\log_{3} (27) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{x} = 27$

Step 2.3

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{3}$

Step 2.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = 3$

Step 2.5

The variable $x$ is equal to $3$ .

$\log_{3} (\frac{x + 4}{x + 2}) = 3$

Step 3

Rewrite $\log_{3} (\frac{x + 4}{x + 2}) = 3$ 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 $b^{y} = x$ .

$3^{3} = \frac{x + 4}{x + 2}$

Step 4

Cross multiply to remove the fraction.

$x + 4 = 3^{3} (x + 2)$

Step 5

Simplify $3^{3} (x + 2)$ .

Tap for more steps...

Step 5.1

Raise $3$ to the power of $3$ .

$x + 4 = 27 (x + 2)$

Step 5.2

Apply the distributive property.

$x + 4 = 27 x + 27 \cdot 2$

Step 5.3

Multiply $27$ by $2$ .

$x + 4 = 27 x + 54$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $27 x$ from both sides of the equation.

$x + 4 - 27 x = 54$

Step 6.2

Subtract $27 x$ from $x$ .

$- 26 x + 4 = 54$

Step 7

Factor $2$ out of $- 26 x + 4$ .

Tap for more steps...

Step 7.1

Factor $2$ out of $- 26 x$ .

$2 (- 13 x) + 4 = 54$

Step 7.2

Factor $2$ out of $4$ .

$2 (- 13 x) + 2 (2) = 54$

Step 7.3

Factor $2$ out of $2 (- 13 x) + 2 (2)$ .

$2 (- 13 x + 2) = 54$

Step 8

Divide each term in $2 (- 13 x + 2) = 54$ by $2$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $2 (- 13 x + 2) = 54$ by $2$ .

$\frac{2 (- 13 x + 2)}{2} = \frac{54}{2}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{2 (- 13 x + 2)}{2} = \frac{54}{2}$

Step 8.2.1.2

Divide $- 13 x + 2$ by $1$ .

$- 13 x + 2 = \frac{54}{2}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Divide $54$ by $2$ .

$- 13 x + 2 = 27$

Step 9

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

Tap for more steps...

Step 9.1

Subtract $2$ from both sides of the equation.

$- 13 x = 27 - 2$

Step 9.2

Subtract $2$ from $27$ .

$- 13 x = 25$

Step 10

Divide each term in $- 13 x = 25$ by $- 13$ and simplify.

Tap for more steps...

Step 10.1

Divide each term in $- 13 x = 25$ by $- 13$ .

$\frac{- 13 x}{- 13} = \frac{25}{- 13}$

Step 10.2

Simplify the left side.

Tap for more steps...

Step 10.2.1

Cancel the common factor of $- 13$ .

Tap for more steps...

Step 10.2.1.1

Cancel the common factor.

$\frac{- 13 x}{- 13} = \frac{25}{- 13}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{25}{- 13}$

Step 10.3

Simplify the right side.

Tap for more steps...

Step 10.3.1

Move the negative in front of the fraction.

$x = - \frac{25}{13}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{25}{13}$

Decimal Form:

$x = - 1.\overline{923076}$

Mixed Number Form:

$x = - 1 \frac{12}{13}$