Finite Math Examples

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

Step 1

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

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

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

Simplify $5 \log_{2} (x)$ by moving $5$ inside the logarithm.

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

Step 2.1.2

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

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

Step 2.1.3

Reduce the expression $\frac{x^{5}}{2 x^{3}}$ by cancelling the common factors.

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Step 2.1.3.1

Factor $x^{3}$ out of $x^{5}$ .

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

Step 2.1.3.2

Factor $x^{3}$ out of $2 x^{3}$ .

$\log_{2} (\frac{x^{3} x^{2}}{x^{3} \cdot 2}) = 5$

Step 2.1.3.3

Cancel the common factor.

$\log_{2} (\frac{x^{3} x^{2}}{x^{3} \cdot 2}) = 5$

Step 2.1.3.4

Rewrite the expression.

$\log_{2} (\frac{x^{2}}{2}) = 5$

Step 2.1.4

Rewrite $\frac{x^{2}}{2}$ as $x^{2} \cdot 2^{- 1}$ .

$\log_{2} (x^{2} \cdot 2^{- 1}) = 5$

Step 2.1.5

Rewrite $\log_{2} (x^{2} \cdot 2^{- 1})$ as $\log_{2} (x^{2}) + \log_{2} (2^{- 1})$ .

$\log_{2} (x^{2}) + \log_{2} (2^{- 1}) = 5$

Step 2.1.6

Use logarithm rules to move $- 1$ out of the exponent.

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

Step 2.1.7

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

$\log_{2} (x^{2}) - 1 \cdot 1 = 5$

Step 2.1.8

Multiply $- 1$ by $1$ .

$\log_{2} (x^{2}) - 1 = 5$

Step 3

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

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Step 3.1

Add $1$ to both sides of the equation.

$\log_{2} (x^{2}) = 5 + 1$

Step 3.2

Add $5$ and $1$ .

$\log_{2} (x^{2}) = 6$

Step 4

Write in exponential form.

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Step 4.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 2$ , $x = x^{2}$ , and $y = 6$ .

$b = 2$

$x = x^{2}$

$y = 6$

Step 4.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$2^{6} = x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $x^{2} = 2^{6}$ .

$x^{2} = 2^{6}$

Step 5.2

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

$x = \pm \sqrt{2^{6}}$

Step 5.3

Simplify $\pm \sqrt{2^{6}}$ .

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Step 5.3.1

Raise $2$ to the power of $6$ .

$x = \pm \sqrt{64}$

Step 5.3.2

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

Step 5.3.3

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

$x = \pm 8$

Step 5.4

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

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Step 5.4.1

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

$x = 8$

Step 5.4.2

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

$x = - 8$

Step 5.4.3

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

$x = 8, - 8$

Step 6

Exclude the solutions that do not make $5 \log_{2} (x) - \log_{2} (2 x^{3}) = 5$ true.

$x = 8$