Algebra Examples

$\frac{3}{2} \log_{27} (3) + \log_{7} (49) - \frac{3}{2} \log_{6} (\sqrt[3]{6})$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $27$ of $3$ is $\frac{1}{3}$ .

$\frac{3}{2} \cdot \frac{1}{3} + \log_{7} (49) - \frac{3}{2} \cdot \log_{6} (\sqrt[3]{6})$

Step 1.2

Cancel the common factor of $3$ .

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Step 1.2.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{1}{3} + \log_{7} (49) - \frac{3}{2} \cdot \log_{6} (\sqrt[3]{6})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{2} + \log_{7} (49) - \frac{3}{2} \cdot \log_{6} (\sqrt[3]{6})$

Step 1.3

Logarithm base $7$ of $49$ is $2$ .

$\frac{1}{2} + 2 - \frac{3}{2} \cdot \log_{6} (\sqrt[3]{6})$

Step 1.4

Multiply $- \frac{3}{2} \log_{6} (\sqrt[3]{6})$ .

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Step 1.4.1

Reorder $\log_{6} (\sqrt[3]{6})$ and $\frac{3}{2}$ .

$\frac{1}{2} + 2 - (\frac{3}{2} \log_{6} (\sqrt[3]{6}))$

Step 1.4.2

Simplify $\frac{3}{2} \log_{6} (\sqrt[3]{6})$ by moving $\frac{3}{2}$ inside the logarithm.

$\frac{1}{2} + 2 - \log_{6} ({\sqrt[3]{6}}^{\frac{3}{2}})$

Step 1.5

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

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Step 1.5.1

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

$\frac{1}{2} + 2 - \log_{6} ({(6^{\frac{1}{3}})}^{\frac{3}{2}})$

Step 1.5.2

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

$\frac{1}{2} + 2 - \log_{6} (6^{\frac{1}{3} \cdot \frac{3}{2}})$

Step 1.5.3

Multiply $\frac{1}{3}$ by $\frac{3}{2}$ .

$\frac{1}{2} + 2 - \log_{6} (6^{\frac{3}{3 \cdot 2}})$

Step 1.5.4

Multiply $3$ by $2$ .

$\frac{1}{2} + 2 - \log_{6} (6^{\frac{3}{6}})$

Step 1.5.5

Cancel the common factor of $3$ and $6$ .

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Step 1.5.5.1

Factor $3$ out of $3$ .

$\frac{1}{2} + 2 - \log_{6} (6^{\frac{3 (1)}{6}})$

Step 1.5.5.2

Cancel the common factors.

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Step 1.5.5.2.1

Factor $3$ out of $6$ .

$\frac{1}{2} + 2 - \log_{6} (6^{\frac{3 \cdot 1}{3 \cdot 2}})$

Step 1.5.5.2.2

Cancel the common factor.

$\frac{1}{2} + 2 - \log_{6} (6^{\frac{3 \cdot 1}{3 \cdot 2}})$

Step 1.5.5.2.3

Rewrite the expression.

$\frac{1}{2} + 2 - \log_{6} (6^{\frac{1}{2}})$

Step 1.5.6

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

$\frac{1}{2} + 2 - \log_{6} (\sqrt{6})$

Step 2

Find the common denominator.

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Step 2.1

Write $2$ as a fraction with denominator $1$ .

$\frac{1}{2} + \frac{2}{1} - \log_{6} (\sqrt{6})$

Step 2.2

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

$\frac{1}{2} + \frac{2}{1} \cdot \frac{2}{2} - \log_{6} (\sqrt{6})$

Step 2.3

Multiply $\frac{2}{1}$ by $\frac{2}{2}$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2} - \log_{6} (\sqrt{6})$

Step 2.4

Write $- \log_{6} (\sqrt{6})$ as a fraction with denominator $1$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2} + \frac{- \log_{6} (\sqrt{6})}{1}$

Step 2.5

Multiply $\frac{- \log_{6} (\sqrt{6})}{1}$ by $\frac{2}{2}$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2} + \frac{- \log_{6} (\sqrt{6})}{1} \cdot \frac{2}{2}$

Step 2.6

Multiply $\frac{- \log_{6} (\sqrt{6})}{1}$ by $\frac{2}{2}$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2} + \frac{- \log_{6} (\sqrt{6}) \cdot 2}{2}$

Step 3

Combine the numerators over the common denominator.

$\frac{1 + 2 \cdot 2 - \log_{6} (\sqrt{6}) \cdot 2}{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply $2$ by $2$ .

$\frac{1 + 4 - \log_{6} (\sqrt{6}) \cdot 2}{2}$

Step 4.2

Multiply $- \log_{6} (\sqrt{6}) \cdot 2$ .

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Step 4.2.1

Multiply $2$ by $- 1$ .

$\frac{1 + 4 - 2 \log_{6} (\sqrt{6})}{2}$

Step 4.2.2

Simplify $- 2 \log_{6} (\sqrt{6})$ by moving $2$ inside the logarithm.

$\frac{1 + 4 - \log_{6} ({\sqrt{6}}^{2})}{2}$

Step 4.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{1 + 4 - \log_{6} ({(6^{\frac{1}{2}})}^{2})}{2}$

Step 4.3.2

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

$\frac{1 + 4 - \log_{6} (6^{\frac{1}{2} \cdot 2})}{2}$

Step 4.3.3

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

$\frac{1 + 4 - \log_{6} (6^{\frac{2}{2}})}{2}$

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{1 + 4 - \log_{6} (6^{\frac{2}{2}})}{2}$

Step 4.3.4.2

Rewrite the expression.

$\frac{1 + 4 - \log_{6} (6^{1})}{2}$

Step 4.3.5

Evaluate the exponent.

$\frac{1 + 4 - \log_{6} (6)}{2}$

Step 4.4

Logarithm base $6$ of $6$ is $1$ .

$\frac{1 + 4 - 1 \cdot 1}{2}$

Step 4.5

Multiply $- 1$ by $1$ .

$\frac{1 + 4 - 1}{2}$

Step 5

Simplify the expression.

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Step 5.1

Add $1$ and $4$ .

$\frac{5 - 1}{2}$

Step 5.2

Subtract $1$ from $5$ .

$\frac{4}{2}$

Step 5.3

Divide $4$ by $2$ .

$2$