Finite Math Examples

$\log_{2} (4 x - 8) = 3 + \log_{2} (2 x)$

Step 1

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

$\log_{2} (4 x - 8) - \log_{2} (2 x) = 3$

Step 2

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

$\log_{2} (\frac{4 x - 8}{2 x}) = 3$

Step 3

Cancel the common factor of $4 x - 8$ and $2$ .

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Step 3.1

Factor $2$ out of $4 x$ .

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

Step 3.2

Factor $2$ out of $- 8$ .

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

Step 3.3

Factor $2$ out of $2 (2 x) + 2 (- 4)$ .

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $2 x$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

Factor $2$ out of $2 x$ .

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

Step 4.2

Factor $2$ out of $- 4$ .

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

Step 4.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

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

Step 5

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

$2^{3} = \frac{2 (x - 2)}{x}$

Step 6

Cross multiply to remove the fraction.

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

Step 7

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

$2 (x - 2) = 8 x$

Step 8

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

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Step 8.1

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

$2 (x - 2) - 8 x = 0$

Step 8.2

Simplify each term.

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Step 8.2.1

Apply the distributive property.

$2 x + 2 \cdot - 2 - 8 x = 0$

Step 8.2.2

Multiply $2$ by $- 2$ .

$2 x - 4 - 8 x = 0$

Step 8.3

Subtract $8 x$ from $2 x$ .

$- 6 x - 4 = 0$

Step 9

Factor $2$ out of $- 6 x - 4$ .

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Step 9.1

Factor $2$ out of $- 6 x$ .

$2 (- 3 x) - 4 = 0$

Step 9.2

Factor $2$ out of $- 4$ .

$2 (- 3 x) + 2 (- 2) = 0$

Step 9.3

Factor $2$ out of $2 (- 3 x) + 2 (- 2)$ .

$2 (- 3 x - 2) = 0$

Step 10

Divide each term in $2 (- 3 x - 2) = 0$ by $2$ and simplify.

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Step 10.1

Divide each term in $2 (- 3 x - 2) = 0$ by $2$ .

$\frac{2 (- 3 x - 2)}{2} = \frac{0}{2}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $2$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{2 (- 3 x - 2)}{2} = \frac{0}{2}$

Step 10.2.1.2

Divide $- 3 x - 2$ by $1$ .

$- 3 x - 2 = \frac{0}{2}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Divide $0$ by $2$ .

$- 3 x - 2 = 0$

Step 11

Add $2$ to both sides of the equation.

$- 3 x = 2$

Step 12

Divide each term in $- 3 x = 2$ by $- 3$ and simplify.

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Step 12.1

Divide each term in $- 3 x = 2$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{2}{- 3}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $- 3$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{2}{- 3}$

Step 12.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{- 3}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Step 13

Exclude the solutions that do not make $\log_{2} (4 x - 8) = 3 + \log_{2} (2 x)$ true.

$No solution$