Evaluate 3/2 log base 27 of 3+ log base 7 of 49-3/2 log base 6 of cube root of 6

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

Step 1

Simplify each term.

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

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

Factor $3$ out of $3$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 2

Find the common denominator.

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

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

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

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

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

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

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

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

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

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

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.

Multiply $2$ by $2$ .

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

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

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Multiply $- 1$ by $1$ .

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

Step 5

Simplify the expression.

Add $1$ and $4$ .

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

Subtract $1$ from $5$ .

$\frac{4}{2}$

Divide $4$ by $2$ .

$2$

Evaluate 3/2 log base 27 of 3+ log base 7 of 49-3/2 log base 6 of cube root of 6

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