Finite Math Examples

$\log_{b} (x) = \frac{2}{3} \cdot \log_{b} (8) + \frac{1}{2} \cdot \log_{b} (16) - \log_{b} (8)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\frac{2}{3} \log_{b} (8) + \frac{1}{2} \log_{b} (16) - \log_{b} (8)$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Combine $\frac{2}{3}$ and $\log_{b} (8)$ .

$\log_{b} (x) = \frac{2 \log_{b} (8)}{3} + \frac{1}{2} \log_{b} (16) - \log_{b} (8)$

Step 1.1.1.2

Combine $\frac{1}{2}$ and $\log_{b} (16)$ .

$\log_{b} (x) = \frac{2 \log_{b} (8)}{3} + \frac{\log_{b} (16)}{2} - \log_{b} (8)$

Step 1.1.2

To write $- \log_{b} (8)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{2 \log_{b} (8)}{3} - \log_{b} (8) \cdot \frac{3}{3}$

Step 1.1.3

Simplify terms.

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Step 1.1.3.1

Combine $- \log_{b} (8)$ and $\frac{3}{3}$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{2 \log_{b} (8)}{3} + \frac{- \log_{b} (8) \cdot 3}{3}$

Step 1.1.3.2

Combine the numerators over the common denominator.

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{2 \log_{b} (8) - \log_{b} (8) \cdot 3}{3}$

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

Simplify the numerator.

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Step 1.1.4.1.1

Factor $\log_{b} (8)$ out of $2 \log_{b} (8) - \log_{b} (8) \cdot 3$ .

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Step 1.1.4.1.1.1

Factor $\log_{b} (8)$ out of $2 \log_{b} (8)$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{\log_{b} (8) \cdot 2 - \log_{b} (8) \cdot 3}{3}$

Step 1.1.4.1.1.2

Factor $\log_{b} (8)$ out of $- \log_{b} (8) \cdot 3$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{\log_{b} (8) \cdot 2 + \log_{b} (8) (- 1 \cdot 3)}{3}$

Step 1.1.4.1.1.3

Factor $\log_{b} (8)$ out of $\log_{b} (8) \cdot 2 + \log_{b} (8) (- 1 \cdot 3)$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{\log_{b} (8) (2 - 1 \cdot 3)}{3}$

Step 1.1.4.1.2

Multiply $- 1$ by $3$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{\log_{b} (8) (2 - 3)}{3}$

Step 1.1.4.1.3

Subtract $3$ from $2$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{\log_{b} (8) \cdot - 1}{3}$

Step 1.1.4.2

Move $- 1$ to the left of $\log_{b} (8)$ .

$\log_{b} (x) = \frac{\log_{b} (16)}{2} + \frac{- 1 \cdot \log_{b} (8)}{3}$

Step 1.1.4.3

Move the negative in front of the fraction.

$\log_{b} (x) = \frac{\log_{b} (16)}{2} - \frac{\log_{b} (8)}{3}$

Step 2

Multiply each term in $\log_{b} (x) = \frac{\log_{b} (16)}{2} - \frac{\log_{b} (8)}{3}$ by $6$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\log_{b} (x) = \frac{\log_{b} (16)}{2} - \frac{\log_{b} (8)}{3}$ by $6$ .

$\log_{b} (x) \cdot 6 = \frac{\log_{b} (16)}{2} \cdot 6 - \frac{\log_{b} (8)}{3} \cdot 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Move $6$ to the left of $\log_{b} (x)$ .

$6 \log_{b} (x) = \frac{\log_{b} (16)}{2} \cdot 6 - \frac{\log_{b} (8)}{3} \cdot 6$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1

Factor $2$ out of $6$ .

$6 \log_{b} (x) = \frac{\log_{b} (16)}{2} \cdot (2 (3)) - \frac{\log_{b} (8)}{3} \cdot 6$

Step 2.3.1.1.2

Cancel the common factor.

$6 \log_{b} (x) = \frac{\log_{b} (16)}{2} \cdot (2 \cdot 3) - \frac{\log_{b} (8)}{3} \cdot 6$

Step 2.3.1.1.3

Rewrite the expression.

$6 \log_{b} (x) = \log_{b} (16) \cdot 3 - \frac{\log_{b} (8)}{3} \cdot 6$

Step 2.3.1.2

Move $3$ to the left of $\log_{b} (16)$ .

$6 \log_{b} (x) = 3 \cdot \log_{b} (16) - \frac{\log_{b} (8)}{3} \cdot 6$

Step 2.3.1.3

Cancel the common factor of $3$ .

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Step 2.3.1.3.1

Move the leading negative in $- \frac{\log_{b} (8)}{3}$ into the numerator.

$6 \log_{b} (x) = 3 \log_{b} (16) + \frac{- \log_{b} (8)}{3} \cdot 6$

Step 2.3.1.3.2

Factor $3$ out of $6$ .

$6 \log_{b} (x) = 3 \log_{b} (16) + \frac{- \log_{b} (8)}{3} \cdot (3 (2))$

Step 2.3.1.3.3

Cancel the common factor.

$6 \log_{b} (x) = 3 \log_{b} (16) + \frac{- \log_{b} (8)}{3} \cdot (3 \cdot 2)$

Step 2.3.1.3.4

Rewrite the expression.

$6 \log_{b} (x) = 3 \log_{b} (16) - \log_{b} (8) \cdot 2$

Step 2.3.1.4

Multiply $2$ by $- 1$ .

$6 \log_{b} (x) = 3 \log_{b} (16) - 2 \log_{b} (8)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $6 \log_{b} (x)$ by moving $6$ inside the logarithm.

$\log_{b} (x^{6}) = 3 \log_{b} (16) - 2 \log_{b} (8)$

Step 4

Simplify the right side.

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Step 4.1

Simplify $3 \log_{b} (16) - 2 \log_{b} (8)$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Simplify $3 \log_{b} (16)$ by moving $3$ inside the logarithm.

$\log_{b} (x^{6}) = \log_{b} (16^{3}) - 2 \log_{b} (8)$

Step 4.1.1.2

Raise $16$ to the power of $3$ .

$\log_{b} (x^{6}) = \log_{b} (4096) - 2 \log_{b} (8)$

Step 4.1.1.3

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

$\log_{b} (x^{6}) = \log_{b} (4096) - \log_{b} (8^{2})$

Step 4.1.1.4

Raise $8$ to the power of $2$ .

$\log_{b} (x^{6}) = \log_{b} (4096) - \log_{b} (64)$

Step 4.1.2

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

$\log_{b} (x^{6}) = \log_{b} (\frac{4096}{64})$

Step 4.1.3

Divide $4096$ by $64$ .

$\log_{b} (x^{6}) = \log_{b} (64)$

Step 5

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

$x^{6} = 64$

Step 6

Solve for $x$ .

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Step 6.1

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

$x = \pm \sqrt[6]{64}$

Step 6.2

Simplify $\pm \sqrt[6]{64}$ .

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Step 6.2.1

Rewrite $64$ as $2^{6}$ .

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

Step 6.2.2

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

$x = \pm 2$

Step 6.3

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

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Step 6.3.1

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

$x = 2$

Step 6.3.2

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

$x = - 2$

Step 6.3.3

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

$x = 2, - 2$