Precalculus Examples

$2 \log_{2} (x - 8) + \log_{2} (2) = 3$

Step 1

Subtract $3$ from both sides of the equation.

$2 \log_{2} (x - 8) + \log_{2} (2) - 3 = 0$

Step 2

Simplify $2 \log_{2} (x - 8) + \log_{2} (2) - 3$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Simplify $2 \log_{2} (x - 8)$ by moving $2$ inside the logarithm.

$\log_{2} ({(x - 8)}^{2}) + \log_{2} (2) - 3 = 0$

Step 2.1.2

Logarithm base $2$ of $2$ is $1$ .

$\log_{2} ({(x - 8)}^{2}) + 1 - 3 = 0$

Step 2.2

Subtract $3$ from $1$ .

$\log_{2} ({(x - 8)}^{2}) - 2 = 0$

Step 3

Add $2$ to both sides of the equation.

$\log_{2} ({(x - 8)}^{2}) = 2$

Step 4

Rewrite $\log_{2} ({(x - 8)}^{2}) = 2$ 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 $b^{y} = x$ .

$2^{2} = {(x - 8)}^{2}$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Rewrite the equation as ${(x - 8)}^{2} = 2^{2}$ .

${(x - 8)}^{2} = 2^{2}$

Step 5.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x - 8 | = | 2 |$

Step 5.3

Solve for $x$ .

Tap for more steps...

Step 5.3.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 8 = \pm | 2 |$

Step 5.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$x - 8 = \pm 2$

Step 5.3.3

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

Tap for more steps...

Step 5.3.3.1

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

$x - 8 = 2$

Step 5.3.3.2

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

Tap for more steps...

Step 5.3.3.2.1

Add $8$ to both sides of the equation.

$x = 2 + 8$

Step 5.3.3.2.2

Add $2$ and $8$ .

$x = 10$

Step 5.3.3.3

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

$x - 8 = - 2$

Step 5.3.3.4

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

Tap for more steps...

Step 5.3.3.4.1

Add $8$ to both sides of the equation.

$x = - 2 + 8$

Step 5.3.3.4.2

Add $- 2$ and $8$ .

$x = 6$

Step 5.3.3.5

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

$x = 10, 6$