Finite Math Examples

$\log (x) = \frac{1}{3} \cdot \log (a) + 4 \log (b) - 2 \log (c) - \frac{1}{2} \cdot \log (a - b)$

Step 1

Simplify the right side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{3}$ and $\log (a)$ .

$\log (x) = \frac{\log (a)}{3} + 4 \log (b) - 2 \log (c) - \frac{1}{2} \log (a - b)$

Step 1.1.2

Combine $\log (a - b)$ and $\frac{1}{2}$ .

$\log (x) = \frac{\log (a)}{3} + 4 \log (b) - 2 \log (c) - \frac{\log (a - b)}{2}$

Step 2

Multiply each term in $\log (x) = \frac{\log (a)}{3} + 4 \log (b) - 2 \log (c) - \frac{\log (a - b)}{2}$ by $6$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\log (x) = \frac{\log (a)}{3} + 4 \log (b) - 2 \log (c) - \frac{\log (a - b)}{2}$ by $6$ .

$\log (x) \cdot 6 = \frac{\log (a)}{3} \cdot 6 + 4 \log (b) \cdot 6 - 2 \log (c) \cdot 6 - \frac{\log (a - b)}{2} \cdot 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Move $6$ to the left of $\log (x)$ .

$6 \log (x) = \frac{\log (a)}{3} \cdot 6 + 4 \log (b) \cdot 6 - 2 \log (c) \cdot 6 - \frac{\log (a - b)}{2} \cdot 6$

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 $3$ .

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Step 2.3.1.1.1

Factor $3$ out of $6$ .

$6 \log (x) = \frac{\log (a)}{3} \cdot (3 (2)) + 4 \log (b) \cdot 6 - 2 \log (c) \cdot 6 - \frac{\log (a - b)}{2} \cdot 6$

Step 2.3.1.1.2

Cancel the common factor.

$6 \log (x) = \frac{\log (a)}{3} \cdot (3 \cdot 2) + 4 \log (b) \cdot 6 - 2 \log (c) \cdot 6 - \frac{\log (a - b)}{2} \cdot 6$

Step 2.3.1.1.3

Rewrite the expression.

$6 \log (x) = \log (a) \cdot 2 + 4 \log (b) \cdot 6 - 2 \log (c) \cdot 6 - \frac{\log (a - b)}{2} \cdot 6$

Step 2.3.1.2

Move $2$ to the left of $\log (a)$ .

$6 \log (x) = 2 \cdot \log (a) + 4 \log (b) \cdot 6 - 2 \log (c) \cdot 6 - \frac{\log (a - b)}{2} \cdot 6$

Step 2.3.1.3

Multiply $6$ by $4$ .

$6 \log (x) = 2 \log (a) + 24 \log (b) - 2 \log (c) \cdot 6 - \frac{\log (a - b)}{2} \cdot 6$

Step 2.3.1.4

Multiply $6$ by $- 2$ .

$6 \log (x) = 2 \log (a) + 24 \log (b) - 12 \log (c) - \frac{\log (a - b)}{2} \cdot 6$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Move the leading negative in $- \frac{\log (a - b)}{2}$ into the numerator.

$6 \log (x) = 2 \log (a) + 24 \log (b) - 12 \log (c) + \frac{- \log (a - b)}{2} \cdot 6$

Step 2.3.1.5.2

Factor $2$ out of $6$ .

$6 \log (x) = 2 \log (a) + 24 \log (b) - 12 \log (c) + \frac{- \log (a - b)}{2} \cdot (2 (3))$

Step 2.3.1.5.3

Cancel the common factor.

$6 \log (x) = 2 \log (a) + 24 \log (b) - 12 \log (c) + \frac{- \log (a - b)}{2} \cdot (2 \cdot 3)$

Step 2.3.1.5.4

Rewrite the expression.

$6 \log (x) = 2 \log (a) + 24 \log (b) - 12 \log (c) - \log (a - b) \cdot 3$

Step 2.3.1.6

Multiply $3$ by $- 1$ .

$6 \log (x) = 2 \log (a) + 24 \log (b) - 12 \log (c) - 3 \log (a - b)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $6 \log (x)$ by moving $6$ inside the logarithm.

$\log (x^{6}) = 2 \log (a) + 24 \log (b) - 12 \log (c) - 3 \log (a - b)$

Step 4

Simplify the right side.

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Step 4.1

Simplify $2 \log (a) + 24 \log (b) - 12 \log (c) - 3 \log (a - b)$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Simplify $2 \log (a)$ by moving $2$ inside the logarithm.

$\log (x^{6}) = \log (a^{2}) + 24 \log (b) - 12 \log (c) - 3 \log (a - b)$

Step 4.1.1.2

Simplify $24 \log (b)$ by moving $24$ inside the logarithm.

$\log (x^{6}) = \log (a^{2}) + \log (b^{24}) - 12 \log (c) - 3 \log (a - b)$

Step 4.1.1.3

Simplify $- 12 \log (c)$ by moving $12$ inside the logarithm.

$\log (x^{6}) = \log (a^{2}) + \log (b^{24}) - \log (c^{12}) - 3 \log (a - b)$

Step 4.1.1.4

Simplify $- 3 \log (a - b)$ by moving $3$ inside the logarithm.

$\log (x^{6}) = \log (a^{2}) + \log (b^{24}) - \log (c^{12}) - \log ({(a - b)}^{3})$

Step 4.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (x^{6}) = \log (a^{2} b^{24}) - \log (c^{12}) - \log ({(a - b)}^{3})$

Step 4.1.3

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

$\log (x^{6}) = \log (\frac{a^{2} b^{24}}{c^{12}}) - \log ({(a - b)}^{3})$

Step 4.1.4

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

$\log (x^{6}) = \log (\frac{\frac{a^{2} b^{24}}{c^{12}}}{{(a - b)}^{3}})$

Step 4.1.5

Multiply the numerator by the reciprocal of the denominator.

$\log (x^{6}) = \log (\frac{a^{2} b^{24}}{c^{12}} \cdot \frac{1}{{(a - b)}^{3}})$

Step 4.1.6

Combine.

$\log (x^{6}) = \log (\frac{a^{2} b^{24} \cdot 1}{c^{12} {(a - b)}^{3}})$

Step 4.1.7

Multiply $a^{2}$ by $1$ .

$\log (x^{6}) = \log (\frac{a^{2} b^{24}}{c^{12} {(a - b)}^{3}})$

Step 5

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

$x^{6} = \frac{a^{2} b^{24}}{c^{12} {(a - b)}^{3}}$

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.

$x = \pm \sqrt[6]{\frac{a^{2} b^{24}}{c^{12} {(a - b)}^{3}}}$

Step 6.2

Simplify $\pm \sqrt[6]{\frac{a^{2} b^{24}}{c^{12} {(a - b)}^{3}}}$ .

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Step 6.2.1

Rewrite $\frac{a^{2} b^{24}}{c^{12} {(a - b)}^{3}}$ as ${(\frac{b^{4}}{c^{2}})}^{6} \frac{a^{2}}{{(a - b)}^{3}}$ .

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Step 6.2.1.1

Factor the perfect power ${(b^{4})}^{6}$ out of $a^{2} b^{24}$ .

$x = \pm \sqrt[6]{\frac{{(b^{4})}^{6} a^{2}}{c^{12} {(a - b)}^{3}}}$

Step 6.2.1.2

Factor the perfect power ${(c^{2})}^{6}$ out of $c^{12} {(a - b)}^{3}$ .

$x = \pm \sqrt[6]{\frac{{(b^{4})}^{6} a^{2}}{{(c^{2})}^{6} {(a - b)}^{3}}}$

Step 6.2.1.3

Rearrange the fraction $\frac{{(b^{4})}^{6} a^{2}}{{(c^{2})}^{6} {(a - b)}^{3}}$ .

$x = \pm \sqrt[6]{{(\frac{b^{4}}{c^{2}})}^{6} \frac{a^{2}}{{(a - b)}^{3}}}$

Step 6.2.2

Pull terms out from under the radical.

$x = \pm \frac{b^{4}}{c^{2}} \sqrt[6]{\frac{a^{2}}{{(a - b)}^{3}}}$

Step 6.2.3

Rewrite $\sqrt[6]{\frac{a^{2}}{{(a - b)}^{3}}}$ as $\frac{\sqrt[6]{a^{2}}}{\sqrt[6]{{(a - b)}^{3}}}$ .

$x = \pm \frac{b^{4}}{c^{2}} \cdot \frac{\sqrt[6]{a^{2}}}{\sqrt[6]{{(a - b)}^{3}}}$

Step 6.2.4

Combine.

$x = \pm \frac{b^{4} \sqrt[6]{a^{2}}}{c^{2} \sqrt[6]{{(a - b)}^{3}}}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Rewrite $\sqrt[6]{a^{2}}$ as $\sqrt[3]{\sqrt{a^{2}}}$ .

$x = \pm \frac{b^{4} \sqrt[3]{\sqrt{a^{2}}}}{c^{2} \sqrt[6]{{(a - b)}^{3}}}$

Step 6.2.5.2

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

$x = \pm \frac{b^{4} \sqrt[3]{a}}{c^{2} \sqrt[6]{{(a - b)}^{3}}}$

Step 6.2.6

Simplify the denominator.

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Step 6.2.6.1

Rewrite $\sqrt[6]{{(a - b)}^{3}}$ as $\sqrt{\sqrt[3]{{(a - b)}^{3}}}$ .

$x = \pm \frac{b^{4} \sqrt[3]{a}}{c^{2} \sqrt{\sqrt[3]{{(a - b)}^{3}}}}$

Step 6.2.6.2

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

$x = \pm \frac{b^{4} \sqrt[3]{a}}{c^{2} \sqrt{a - b}}$

Step 6.2.7

Multiply $\frac{b^{4} \sqrt[3]{a}}{c^{2} \sqrt{a - b}}$ by $\frac{\sqrt{a - b}}{\sqrt{a - b}}$ .

$x = \pm \frac{b^{4} \sqrt[3]{a}}{c^{2} \sqrt{a - b}} \cdot \frac{\sqrt{a - b}}{\sqrt{a - b}}$

Step 6.2.8

Combine and simplify the denominator.

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Step 6.2.8.1

Multiply $\frac{b^{4} \sqrt[3]{a}}{c^{2} \sqrt{a - b}}$ by $\frac{\sqrt{a - b}}{\sqrt{a - b}}$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} \sqrt{a - b} \sqrt{a - b}}$

Step 6.2.8.2

Move $\sqrt{a - b}$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} (\sqrt{a - b} \sqrt{a - b})}$

Step 6.2.8.3

Raise $\sqrt{a - b}$ to the power of $1$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} ({\sqrt{a - b}}^{1} \sqrt{a - b})}$

Step 6.2.8.4

Raise $\sqrt{a - b}$ to the power of $1$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} ({\sqrt{a - b}}^{1} {\sqrt{a - b}}^{1})}$

Step 6.2.8.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} {\sqrt{a - b}}^{1 + 1}}$

Step 6.2.8.6

Add $1$ and $1$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} {\sqrt{a - b}}^{2}}$

Step 6.2.8.7

Rewrite ${\sqrt{a - b}}^{2}$ as $a - b$ .

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Step 6.2.8.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - b}$ as ${(a - b)}^{\frac{1}{2}}$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} {({(a - b)}^{\frac{1}{2}})}^{2}}$

Step 6.2.8.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} {(a - b)}^{\frac{1}{2} \cdot 2}}$

Step 6.2.8.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} {(a - b)}^{\frac{2}{2}}}$

Step 6.2.8.7.4

Cancel the common factor of $2$ .

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Step 6.2.8.7.4.1

Cancel the common factor.

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} {(a - b)}^{\frac{2}{2}}}$

Step 6.2.8.7.4.2

Rewrite the expression.

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} {(a - b)}^{1}}$

Step 6.2.8.7.5

Simplify.

$x = \pm \frac{b^{4} \sqrt[3]{a} \sqrt{a - b}}{c^{2} (a - b)}$

Step 6.2.9

Simplify the numerator.

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Step 6.2.9.1

Rewrite the expression using the least common index of $6$ .

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Step 6.2.9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - b}$ as ${(a - b)}^{\frac{1}{2}}$ .

$x = \pm \frac{b^{4} ({(a - b)}^{\frac{1}{2}} \sqrt[3]{a})}{c^{2} (a - b)}$

Step 6.2.9.1.2

Rewrite ${(a - b)}^{\frac{1}{2}}$ as ${(a - b)}^{\frac{3}{6}}$ .

$x = \pm \frac{b^{4} ({(a - b)}^{\frac{3}{6}} \sqrt[3]{a})}{c^{2} (a - b)}$

Step 6.2.9.1.3

Rewrite ${(a - b)}^{\frac{3}{6}}$ as $\sqrt[6]{{(a - b)}^{3}}$ .

$x = \pm \frac{b^{4} (\sqrt[6]{{(a - b)}^{3}} \sqrt[3]{a})}{c^{2} (a - b)}$

Step 6.2.9.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{a}$ as $a^{\frac{1}{3}}$ .

$x = \pm \frac{b^{4} (\sqrt[6]{{(a - b)}^{3}} a^{\frac{1}{3}})}{c^{2} (a - b)}$

Step 6.2.9.1.5

Rewrite $a^{\frac{1}{3}}$ as $a^{\frac{2}{6}}$ .

$x = \pm \frac{b^{4} (\sqrt[6]{{(a - b)}^{3}} a^{\frac{2}{6}})}{c^{2} (a - b)}$

Step 6.2.9.1.6

Rewrite $a^{\frac{2}{6}}$ as $\sqrt[6]{a^{2}}$ .

$x = \pm \frac{b^{4} (\sqrt[6]{{(a - b)}^{3}} \sqrt[6]{a^{2}})}{c^{2} (a - b)}$

Step 6.2.9.2

Combine using the product rule for radicals.

$x = \pm \frac{b^{4} \sqrt[6]{{(a - b)}^{3} a^{2}}}{c^{2} (a - b)}$

Step 6.2.10

Reorder factors in $\pm \frac{b^{4} \sqrt[6]{{(a - b)}^{3} a^{2}}}{c^{2} (a - b)}$ .

$x = \pm \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

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.

$x = \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

Step 6.3.2

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

$x = - \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

Step 6.3.3

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

$x = \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

$x = - \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

$x = \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

$x = - \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

$x = \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$

$x = - \frac{b^{4} \sqrt[6]{a^{2} {(a - b)}^{3}}}{c^{2} (a - b)}$