Algebra Examples

$\log_{2} (4 x) - \log_{2} (x - 5) = \log_{2} (8)$

Step 1

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

$\log_{2} (\frac{4 x}{x - 5}) = \log_{2} (8)$

Step 2

Logarithm base $2$ of $8$ is $3$ .

Tap for more steps...

Step 2.1

Rewrite as an equation.

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

Step 2.2

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

$2^{x} = 8$

Step 2.3

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

$2^{x} = 2^{3}$

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 $3$ .

$\log_{2} (\frac{4 x}{x - 5}) = 3$

Step 3

Rewrite $\log_{2} (\frac{4 x}{x - 5}) = 3$ 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 $b^{y} = x$ .

$2^{3} = \frac{4 x}{x - 5}$

Step 4

Cross multiply to remove the fraction.

$4 x = 2^{3} (x - 5)$

Step 5

Simplify $2^{3} (x - 5)$ .

Tap for more steps...

Step 5.1

Raise $2$ to the power of $3$ .

$4 x = 8 (x - 5)$

Step 5.2

Apply the distributive property.

$4 x = 8 x + 8 \cdot - 5$

Step 5.3

Multiply $8$ by $- 5$ .

$4 x = 8 x - 40$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $8 x$ from both sides of the equation.

$4 x - 8 x = - 40$

Step 6.2

Subtract $8 x$ from $4 x$ .

$- 4 x = - 40$

Step 7

Divide each term in $- 4 x = - 40$ by $- 4$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $- 4 x = - 40$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 40}{- 4}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 40}{- 4}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 40}{- 4}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Divide $- 40$ by $- 4$ .

$x = 10$