Precalculus Examples

$2 \log_{3} (x + 4) = \log_{3} (9) + 2$

Step 1

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

$2 \log_{3} (x + 4) - \log_{3} (9) = 2$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Logarithm base $3$ of $9$ is $2$ .

$2 \log_{3} (x + 4) - 1 \cdot 2 = 2$

Step 2.2

Multiply $- 1$ by $2$ .

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

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $2 \log_{3} (x + 4)$ by moving $2$ inside the logarithm.

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

Step 4

Move all terms not containing $\log_{3} ({(x + 4)}^{2})$ to the right side of the equation.

Tap for more steps...

Step 4.1

Add $2$ to both sides of the equation.

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

Step 4.2

Add $2$ and $2$ .

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

Step 5

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

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

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Rewrite the equation as ${(x + 4)}^{2} = 3^{4}$ .

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

Step 6.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 4 = \pm \sqrt{3^{4}}$

Step 6.3

Simplify $\pm \sqrt{3^{4}}$ .

Tap for more steps...

Step 6.3.1

Raise $3$ to the power of $4$ .

$x + 4 = \pm \sqrt{81}$

Step 6.3.2

Rewrite $81$ as $9^{2}$ .

$x + 4 = \pm \sqrt{9^{2}}$

Step 6.3.3

Pull terms out from under the radical, assuming positive real numbers.

$x + 4 = \pm 9$

Step 6.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 4 = 9$

Step 6.4.2

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

Tap for more steps...

Step 6.4.2.1

Subtract $4$ from both sides of the equation.

$x = 9 - 4$

Step 6.4.2.2

Subtract $4$ from $9$ .

$x = 5$

Step 6.4.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 4 = - 9$

Step 6.4.4

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

Tap for more steps...

Step 6.4.4.1

Subtract $4$ from both sides of the equation.

$x = - 9 - 4$

Step 6.4.4.2

Subtract $4$ from $- 9$ .

$x = - 13$

Step 6.4.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5, - 13$

Step 7

Exclude the solutions that do not make $2 \log_{3} (x + 4) = \log_{3} (9) + 2$ true.

$x = 5$