Algebra Examples

$\sqrt[3]{5 x^{k + 1}} \cdot \sqrt[3]{25 x^{k}} = 5 x^{7}$

Step 1

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

$\ln (\sqrt[3]{5 x^{k + 1}} \cdot \sqrt[3]{25 x^{k}}) = \ln (5 x^{7})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (\sqrt[3]{5 x^{k + 1}} \cdot \sqrt[3]{25 x^{k}})$ as $\ln (\sqrt[3]{5 x^{k + 1}}) + \ln (\sqrt[3]{25 x^{k}})$ .

$\ln (\sqrt[3]{5 x^{k + 1}}) + \ln (\sqrt[3]{25 x^{k}}) = \ln (5 x^{7})$

Step 2.2

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

$\ln ({(5 x^{k + 1})}^{\frac{1}{3}}) + \ln (\sqrt[3]{25 x^{k}}) = \ln (5 x^{7})$

Step 2.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25 x^{k}}$ as ${(25 x^{k})}^{\frac{1}{3}}$ .

$\ln ({(5 x^{k + 1})}^{\frac{1}{3}}) + \ln ({(25 x^{k})}^{\frac{1}{3}}) = \ln (5 x^{7})$

Step 2.4

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

$\frac{1}{3} \ln (5 x^{k + 1}) + \ln ({(25 x^{k})}^{\frac{1}{3}}) = \ln (5 x^{7})$

Step 2.5

Expand $\ln ({(25 x^{k})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (5 x^{k + 1}) + \frac{1}{3} \ln (25 x^{k}) = \ln (5 x^{7})$

Step 2.6

Rewrite $\ln (5 x^{k + 1})$ as $\ln (5) + \ln (x^{k + 1})$ .

$\frac{1}{3} (\ln (5) + \ln (x^{k + 1})) + \frac{1}{3} \ln (25 x^{k}) = \ln (5 x^{7})$

Step 2.7

Expand $\ln (x^{k + 1})$ by moving $k + 1$ outside the logarithm.

$\frac{1}{3} (\ln (5) + (k + 1) \ln (x)) + \frac{1}{3} \ln (25 x^{k}) = \ln (5 x^{7})$

Step 2.8

Rewrite $\ln (25 x^{k})$ as $\ln (25) + \ln (x^{k})$ .

$\frac{1}{3} (\ln (5) + (k + 1) \ln (x)) + \frac{1}{3} (\ln (25) + \ln (x^{k})) = \ln (5 x^{7})$

Step 2.9

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

$\frac{1}{3} (\ln (5) + (k + 1) \ln (x)) + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3

Simplify the left side.

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Step 3.1

Simplify $\frac{1}{3} (\ln (5) + (k + 1) \ln (x)) + \frac{1}{3} (\ln (25) + k \ln (x))$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify each term.

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Step 3.1.1.1.1

Apply the distributive property.

$\frac{1}{3} (\ln (5) + k \ln (x) + 1 \ln (x)) + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.1.2

Multiply $\ln (x)$ by $1$ .

$\frac{1}{3} (\ln (5) + k \ln (x) + \ln (x)) + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.2

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

$\frac{1}{3} (k \ln (x) + \ln (5 x)) + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.3

Apply the distributive property.

$\frac{1}{3} (k \ln (x)) + \frac{1}{3} \ln (5 x) + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.4

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

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Step 3.1.1.4.1

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

$\frac{k}{3} \ln (x) + \frac{1}{3} \ln (5 x) + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.4.2

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

$\frac{k \ln (x)}{3} + \frac{1}{3} \ln (5 x) + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.5

Combine $\frac{1}{3}$ and $\ln (5 x)$ .

$\frac{k \ln (x)}{3} + \frac{\ln (5 x)}{3} + \frac{1}{3} (\ln (25) + k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.6

Apply the distributive property.

$\frac{k \ln (x)}{3} + \frac{\ln (5 x)}{3} + \frac{1}{3} \ln (25) + \frac{1}{3} (k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.7

Combine $\frac{1}{3}$ and $\ln (25)$ .

$\frac{k \ln (x)}{3} + \frac{\ln (5 x)}{3} + \frac{\ln (25)}{3} + \frac{1}{3} (k \ln (x)) = \ln (5 x^{7})$

Step 3.1.1.8

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

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Step 3.1.1.8.1

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

$\frac{k \ln (x)}{3} + \frac{\ln (5 x)}{3} + \frac{\ln (25)}{3} + \frac{k}{3} \ln (x) = \ln (5 x^{7})$

Step 3.1.1.8.2

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

$\frac{k \ln (x)}{3} + \frac{\ln (5 x)}{3} + \frac{\ln (25)}{3} + \frac{k \ln (x)}{3} = \ln (5 x^{7})$

Step 3.1.2

Combine the numerators over the common denominator.

$\frac{k \ln (x) + \ln (5 x) + \ln (25) + k \ln (x)}{3} = \ln (5 x^{7})$

Step 3.1.3

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

$\frac{k \ln (x) + \ln (5 x \cdot 25) + k \ln (x)}{3} = \ln (5 x^{7})$

Step 3.1.4

Multiply $25$ by $5$ .

$\frac{k \ln (x) + \ln (125 x) + k \ln (x)}{3} = \ln (5 x^{7})$

Step 3.1.5

Add $k \ln (x)$ and $k \ln (x)$ .

$\frac{2 k \ln (x) + \ln (125 x)}{3} = \ln (5 x^{7})$

Step 3.1.6

Split the fraction $\frac{2 k \ln (x) + \ln (125 x)}{3}$ into two fractions.

$\frac{2 k \ln (x)}{3} + \frac{\ln (125 x)}{3} = \ln (5 x^{7})$

Step 4

Multiply each term in $\frac{2 k \ln (x)}{3} + \frac{\ln (125 x)}{3} = \ln (5 x^{7})$ by $3$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{2 k \ln (x)}{3} + \frac{\ln (125 x)}{3} = \ln (5 x^{7})$ by $3$ .

$\frac{2 k \ln (x)}{3} \cdot 3 + \frac{\ln (125 x)}{3} \cdot 3 = \ln (5 x^{7}) \cdot 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{2 k \ln (x)}{3} \cdot 3 + \frac{\ln (125 x)}{3} \cdot 3 = \ln (5 x^{7}) \cdot 3$

Step 4.2.1.1.2

Rewrite the expression.

$2 k \ln (x) + \frac{\ln (125 x)}{3} \cdot 3 = \ln (5 x^{7}) \cdot 3$

Step 4.2.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.2.1

Cancel the common factor.

$2 k \ln (x) + \frac{\ln (125 x)}{3} \cdot 3 = \ln (5 x^{7}) \cdot 3$

Step 4.2.1.2.2

Rewrite the expression.

$2 k \ln (x) + \ln (125 x) = \ln (5 x^{7}) \cdot 3$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move $3$ to the left of $\ln (5 x^{7})$ .

$2 k \ln (x) + \ln (125 x) = 3 \ln (5 x^{7})$

Step 5

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

$2 k \ln (x) + \ln (125 x) - 3 \ln (5 x^{7}) = 0$

Step 6

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

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Step 6.1

Subtract $\ln (125 x)$ from both sides of the equation.

$2 k \ln (x) - 3 \ln (5 x^{7}) = - \ln (125 x)$

Step 6.2

Add $3 \ln (5 x^{7})$ to both sides of the equation.

$2 k \ln (x) = - \ln (125 x) + 3 \ln (5 x^{7})$

Step 7

Divide each term in $2 k \ln (x) = - \ln (125 x) + 3 \ln (5 x^{7})$ by $2 \ln (x)$ and simplify.

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Step 7.1

Divide each term in $2 k \ln (x) = - \ln (125 x) + 3 \ln (5 x^{7})$ by $2 \ln (x)$ .

$\frac{2 k \ln (x)}{2 \ln (x)} = \frac{- \ln (125 x)}{2 \ln (x)} + \frac{3 \ln (5 x^{7})}{2 \ln (x)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{2 k \ln (x)}{2 \ln (x)} = \frac{- \ln (125 x)}{2 \ln (x)} + \frac{3 \ln (5 x^{7})}{2 \ln (x)}$

Step 7.2.1.2

Rewrite the expression.

$\frac{k \ln (x)}{\ln (x)} = \frac{- \ln (125 x)}{2 \ln (x)} + \frac{3 \ln (5 x^{7})}{2 \ln (x)}$

Step 7.2.2

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

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Step 7.2.2.1

Cancel the common factor.

$\frac{k \ln (x)}{\ln (x)} = \frac{- \ln (125 x)}{2 \ln (x)} + \frac{3 \ln (5 x^{7})}{2 \ln (x)}$

Step 7.2.2.2

Divide $k$ by $1$ .

$k = \frac{- \ln (125 x)}{2 \ln (x)} + \frac{3 \ln (5 x^{7})}{2 \ln (x)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$k = - \frac{\ln (125 x)}{2 \ln (x)} + \frac{3 \ln (5 x^{7})}{2 \ln (x)}$