Finite Math Examples

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

Step 1

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

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

Step 2

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

$\log_{b} (\frac{8 + x}{x}) = \log_{b} (2^{3})$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{8 + x}{x} = 2^{3}$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

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

$\frac{8 + x}{x} = 8$

Step 4.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 4.2.2

The LCM of one and any expression is the expression.

$x$

Step 4.3

Multiply each term in $\frac{8 + x}{x} = 8$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 4.3.1

Multiply each term in $\frac{8 + x}{x} = 8$ by $x$ .

$\frac{8 + x}{x} x = 8 x$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

$\frac{8 + x}{x} x = 8 x$

Step 4.3.2.1.2

Rewrite the expression.

$8 + x = 8 x$

Step 4.4

Solve the equation.

Tap for more steps...

Step 4.4.1

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

Tap for more steps...

Step 4.4.1.1

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

$8 + x - 8 x = 0$

Step 4.4.1.2

Subtract $8 x$ from $x$ .

$8 - 7 x = 0$

Step 4.4.2

Subtract $8$ from both sides of the equation.

$- 7 x = - 8$

Step 4.4.3

Divide each term in $- 7 x = - 8$ by $- 7$ and simplify.

Tap for more steps...

Step 4.4.3.1

Divide each term in $- 7 x = - 8$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 8}{- 7}$

Step 4.4.3.2

Simplify the left side.

Tap for more steps...

Step 4.4.3.2.1

Cancel the common factor of $- 7$ .

Tap for more steps...

Step 4.4.3.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 8}{- 7}$

Step 4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 7}$

Step 4.4.3.3

Simplify the right side.

Tap for more steps...

Step 4.4.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{8}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{7}$

Decimal Form:

$x = 1.\overline{142857}$

Mixed Number Form:

$x = 1 \frac{1}{7}$