Algebra Examples

$\sqrt[3]{162 x^{c} y^{5}} = 3 x^{2} y (\sqrt[3]{6 y^{d}})$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (\sqrt[3]{162 x^{c} y^{5}}) = \ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$

Step 2

Expand the left side.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{162 x^{c} y^{5}}$ as ${(162 x^{c} y^{5})}^{\frac{1}{3}}$ .

$\ln ({(162 x^{c} y^{5})}^{\frac{1}{3}}) = \ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$

Step 2.2

Expand $\ln ({(162 x^{c} y^{5})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (162 x^{c} y^{5}) = \ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$

Step 2.3

Rewrite $\ln (162 x^{c} y^{5})$ as $\ln (162 x^{c}) + \ln (y^{5})$ .

$\frac{1}{3} (\ln (162 x^{c}) + \ln (y^{5})) = \ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$

Step 2.4

Rewrite $\ln (162 x^{c})$ as $\ln (162) + \ln (x^{c})$ .

$\frac{1}{3} (\ln (162) + \ln (x^{c}) + \ln (y^{5})) = \ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$

Step 2.5

Expand $\ln (x^{c})$ by moving $c$ outside the logarithm.

$\frac{1}{3} (\ln (162) + c \ln (x) + \ln (y^{5})) = \ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$

Step 2.6

Expand $\ln (y^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (3 x^{2} y (\sqrt[3]{6 y^{d}}))$ as $\ln (3 x^{2} y) + \ln (\sqrt[3]{6 y^{d}})$ .

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3 x^{2} y) + \ln (\sqrt[3]{6 y^{d}})$

Step 3.2

Rewrite $\ln (3 x^{2} y)$ as $\ln (3 x^{2}) + \ln (y)$ .

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3 x^{2}) + \ln (y) + \ln (\sqrt[3]{6 y^{d}})$

Step 3.3

Rewrite $\ln (3 x^{2})$ as $\ln (3) + \ln (x^{2})$ .

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3) + \ln (x^{2}) + \ln (y) + \ln (\sqrt[3]{6 y^{d}})$

Step 3.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 y^{d}}$ as ${(6 y^{d})}^{\frac{1}{3}}$ .

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3) + \ln (x^{2}) + \ln (y) + \ln ({(6 y^{d})}^{\frac{1}{3}})$

Step 3.5

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \ln ({(6 y^{d})}^{\frac{1}{3}})$

Step 3.6

Expand $\ln ({(6 y^{d})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} \ln (6 y^{d})$

Step 3.7

Rewrite $\ln (6 y^{d})$ as $\ln (6) + \ln (y^{d})$ .

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + \ln (y^{d}))$

Step 3.8

Expand $\ln (y^{d})$ by moving $d$ outside the logarithm.

$\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 4

Simplify the left side.

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Step 4.1

Simplify $\frac{1}{3} (\ln (162) + c \ln (x) + 5 \ln (y))$ .

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Step 4.1.1

Apply the distributive property.

$\frac{1}{3} \ln (162) + \frac{1}{3} (c \ln (x)) + \frac{1}{3} (5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Combine $\frac{1}{3}$ and $\ln (162)$ .

$\frac{\ln (162)}{3} + \frac{1}{3} (c \ln (x)) + \frac{1}{3} (5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 4.1.2.2

Multiply $\frac{1}{3} (c \ln (x))$ .

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Step 4.1.2.2.1

Combine $c$ and $\frac{1}{3}$ .

$\frac{\ln (162)}{3} + \frac{c}{3} \ln (x) + \frac{1}{3} (5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 4.1.2.2.2

Combine $\frac{c}{3}$ and $\ln (x)$ .

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{1}{3} (5 \ln (y)) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 4.1.2.3

Multiply $\frac{1}{3} (5 \ln (y))$ .

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Step 4.1.2.3.1

Combine $5$ and $\frac{1}{3}$ .

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{5}{3} \ln (y) = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 4.1.2.3.2

Combine $\frac{5}{3}$ and $\ln (y)$ .

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{5 \ln (y)}{3} = \ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 5

Simplify the right side.

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Step 5.1

Simplify $\ln (3) + 2 \ln (x) + \ln (y) + \frac{1}{3} (\ln (6) + d \ln (y))$ .

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Step 5.1.1

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

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \ln (3 y) + \frac{1}{3} (\ln (6) + d \ln (y))$

Step 5.1.2

Simplify each term.

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Step 5.1.2.1

Apply the distributive property.

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \ln (3 y) + \frac{1}{3} \ln (6) + \frac{1}{3} (d \ln (y))$

Step 5.1.2.2

Combine $\frac{1}{3}$ and $\ln (6)$ .

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \ln (3 y) + \frac{\ln (6)}{3} + \frac{1}{3} (d \ln (y))$

Step 5.1.2.3

Multiply $\frac{1}{3} (d \ln (y))$ .

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Step 5.1.2.3.1

Combine $d$ and $\frac{1}{3}$ .

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \ln (3 y) + \frac{\ln (6)}{3} + \frac{d}{3} \ln (y)$

Step 5.1.2.3.2

Combine $\frac{d}{3}$ and $\ln (y)$ .

$\frac{\ln (162)}{3} + \frac{c \ln (x)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \ln (3 y) + \frac{\ln (6)}{3} + \frac{d \ln (y)}{3}$

Step 6

Reorder $\frac{\ln (162)}{3}$ and $\frac{c \ln (x)}{3}$ .

$\frac{c \ln (x)}{3} + \frac{\ln (162)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \ln (3 y) + \frac{\ln (6)}{3} + \frac{d \ln (y)}{3}$

Step 7

Simplify the right side.

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Step 7.1

Move $\frac{\ln (6)}{3}$ .

$\frac{c \ln (x)}{3} + \frac{\ln (162)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \ln (3 y) + \frac{d \ln (y)}{3} + \frac{\ln (6)}{3}$

Step 7.2

Move $\ln (3 y)$ .

$\frac{c \ln (x)}{3} + \frac{\ln (162)}{3} + \frac{5 \ln (y)}{3} = 2 \ln (x) + \frac{d \ln (y)}{3} + \ln (3 y) + \frac{\ln (6)}{3}$

Step 7.3

Reorder $2 \ln (x)$ and $\frac{d \ln (y)}{3}$ .

$\frac{c \ln (x)}{3} + \frac{\ln (162)}{3} + \frac{5 \ln (y)}{3} = \frac{d \ln (y)}{3} + 2 \ln (x) + \ln (3 y) + \frac{\ln (6)}{3}$

Step 8

Multiply each term in $\frac{c \ln (x)}{3} + \frac{\ln (162)}{3} + \frac{5 \ln (y)}{3} = \frac{d \ln (y)}{3} + 2 \ln (x) + \ln (3 y) + \frac{\ln (6)}{3}$ by $3$ to eliminate the fractions.

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Step 8.1

Multiply each term in $\frac{c \ln (x)}{3} + \frac{\ln (162)}{3} + \frac{5 \ln (y)}{3} = \frac{d \ln (y)}{3} + 2 \ln (x) + \ln (3 y) + \frac{\ln (6)}{3}$ by $3$ .

$\frac{c \ln (x)}{3} \cdot 3 + \frac{\ln (162)}{3} \cdot 3 + \frac{5 \ln (y)}{3} \cdot 3 = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.2

Simplify the left side.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Cancel the common factor of $3$ .

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Step 8.2.1.1.1

Cancel the common factor.

$\frac{c \ln (x)}{3} \cdot 3 + \frac{\ln (162)}{3} \cdot 3 + \frac{5 \ln (y)}{3} \cdot 3 = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.2.1.1.2

Rewrite the expression.

$c \ln (x) + \frac{\ln (162)}{3} \cdot 3 + \frac{5 \ln (y)}{3} \cdot 3 = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.2.1.2

Cancel the common factor of $3$ .

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Step 8.2.1.2.1

Cancel the common factor.

$c \ln (x) + \frac{\ln (162)}{3} \cdot 3 + \frac{5 \ln (y)}{3} \cdot 3 = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.2.1.2.2

Rewrite the expression.

$c \ln (x) + \ln (162) + \frac{5 \ln (y)}{3} \cdot 3 = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.2.1.3

Cancel the common factor of $3$ .

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Step 8.2.1.3.1

Cancel the common factor.

$c \ln (x) + \ln (162) + \frac{5 \ln (y)}{3} \cdot 3 = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.2.1.3.2

Rewrite the expression.

$c \ln (x) + \ln (162) + 5 \ln (y) = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $3$ .

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Step 8.3.1.1.1

Cancel the common factor.

$c \ln (x) + \ln (162) + 5 \ln (y) = \frac{d \ln (y)}{3} \cdot 3 + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.3.1.1.2

Rewrite the expression.

$c \ln (x) + \ln (162) + 5 \ln (y) = d \ln (y) + 2 \ln (x) \cdot 3 + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.3.1.2

Multiply $3$ by $2$ .

$c \ln (x) + \ln (162) + 5 \ln (y) = d \ln (y) + 6 \ln (x) + \ln (3 y) \cdot 3 + \frac{\ln (6)}{3} \cdot 3$

Step 8.3.1.3

Move $3$ to the left of $\ln (3 y)$ .

$c \ln (x) + \ln (162) + 5 \ln (y) = d \ln (y) + 6 \ln (x) + 3 \cdot \ln (3 y) + \frac{\ln (6)}{3} \cdot 3$

Step 8.3.1.4

Cancel the common factor of $3$ .

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Step 8.3.1.4.1

Cancel the common factor.

$c \ln (x) + \ln (162) + 5 \ln (y) = d \ln (y) + 6 \ln (x) + 3 \ln (3 y) + \frac{\ln (6)}{3} \cdot 3$

Step 8.3.1.4.2

Rewrite the expression.

$c \ln (x) + \ln (162) + 5 \ln (y) = d \ln (y) + 6 \ln (x) + 3 \ln (3 y) + \ln (6)$

Step 9

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

$c \ln (x) + \ln (162) + 5 \ln (y) - d \ln (y) - 6 \ln (x) - 3 \ln (3 y) - \ln (6) = 0$

Step 10

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

$c \ln (x) + \ln (\frac{162}{6}) + 5 \ln (y) - d \ln (y) - 6 \ln (x) - 3 \ln (3 y) = 0$

Step 11

Divide $162$ by $6$ .

$c \ln (x) + \ln (27) + 5 \ln (y) - d \ln (y) - 6 \ln (x) - 3 \ln (3 y) = 0$

Step 12

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

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Step 12.1

Subtract $\ln (27)$ from both sides of the equation.

$c \ln (x) + 5 \ln (y) - d \ln (y) - 6 \ln (x) - 3 \ln (3 y) = - \ln (27)$

Step 12.2

Subtract $5 \ln (y)$ from both sides of the equation.

$c \ln (x) - d \ln (y) - 6 \ln (x) - 3 \ln (3 y) = - \ln (27) - 5 \ln (y)$

Step 12.3

Add $d \ln (y)$ to both sides of the equation.

$c \ln (x) - 6 \ln (x) - 3 \ln (3 y) = - \ln (27) - 5 \ln (y) + d \ln (y)$

Step 12.4

Add $6 \ln (x)$ to both sides of the equation.

$c \ln (x) - 3 \ln (3 y) = - \ln (27) - 5 \ln (y) + d \ln (y) + 6 \ln (x)$

Step 12.5

Add $3 \ln (3 y)$ to both sides of the equation.

$c \ln (x) = - \ln (27) - 5 \ln (y) + d \ln (y) + 6 \ln (x) + 3 \ln (3 y)$

Step 13

Divide each term in $c \ln (x) = - \ln (27) - 5 \ln (y) + d \ln (y) + 6 \ln (x) + 3 \ln (3 y)$ by $\ln (x)$ and simplify.

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Step 13.1

Divide each term in $c \ln (x) = - \ln (27) - 5 \ln (y) + d \ln (y) + 6 \ln (x) + 3 \ln (3 y)$ by $\ln (x)$ .

$\frac{c \ln (x)}{\ln (x)} = \frac{- \ln (27)}{\ln (x)} + \frac{- 5 \ln (y)}{\ln (x)} + \frac{d \ln (y)}{\ln (x)} + \frac{6 \ln (x)}{\ln (x)} + \frac{3 \ln (3 y)}{\ln (x)}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $\ln (x)$ .

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Step 13.2.1.1

Cancel the common factor.

$\frac{c \ln (x)}{\ln (x)} = \frac{- \ln (27)}{\ln (x)} + \frac{- 5 \ln (y)}{\ln (x)} + \frac{d \ln (y)}{\ln (x)} + \frac{6 \ln (x)}{\ln (x)} + \frac{3 \ln (3 y)}{\ln (x)}$

Step 13.2.1.2

Divide $c$ by $1$ .

$c = \frac{- \ln (27)}{\ln (x)} + \frac{- 5 \ln (y)}{\ln (x)} + \frac{d \ln (y)}{\ln (x)} + \frac{6 \ln (x)}{\ln (x)} + \frac{3 \ln (3 y)}{\ln (x)}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Simplify each term.

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Step 13.3.1.1

Move the negative in front of the fraction.

$c = - \frac{\ln (27)}{\ln (x)} + \frac{- 5 \ln (y)}{\ln (x)} + \frac{d \ln (y)}{\ln (x)} + \frac{6 \ln (x)}{\ln (x)} + \frac{3 \ln (3 y)}{\ln (x)}$

Step 13.3.1.2

Move the negative in front of the fraction.

$c = - \frac{\ln (27)}{\ln (x)} - \frac{5 \ln (y)}{\ln (x)} + \frac{d \ln (y)}{\ln (x)} + \frac{6 \ln (x)}{\ln (x)} + \frac{3 \ln (3 y)}{\ln (x)}$

Step 13.3.1.3

Cancel the common factor of $\ln (x)$ .

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Step 13.3.1.3.1

Cancel the common factor.

$c = - \frac{\ln (27)}{\ln (x)} - \frac{5 \ln (y)}{\ln (x)} + \frac{d \ln (y)}{\ln (x)} + \frac{6 \ln (x)}{\ln (x)} + \frac{3 \ln (3 y)}{\ln (x)}$

Step 13.3.1.3.2

Divide $6$ by $1$ .

$c = - \frac{\ln (27)}{\ln (x)} - \frac{5 \ln (y)}{\ln (x)} + \frac{d \ln (y)}{\ln (x)} + 6 + \frac{3 \ln (3 y)}{\ln (x)}$