Write as a Single Logarithm 1/2(4 log of x- log of square root of x^5)+ log of cube root of x^2

$\frac{1}{2} (4 \log (x) - \log (\sqrt{x^{5}})) + \log (\sqrt[3]{x^{2}})$

Step 1

Simplify each term.

Simplify $4 \log (x)$ by moving $4$ inside the logarithm.

$\frac{1}{2} \cdot (\log (x^{4}) - \log (\sqrt{x^{5}})) + \log (\sqrt[3]{x^{2}})$

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

Factor out $x^{4}$ .

$\frac{1}{2} \cdot (\log (x^{4}) - \log (\sqrt{x^{4} x})) + \log (\sqrt[3]{x^{2}})$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{1}{2} \cdot (\log (x^{4}) - \log (\sqrt{{(x^{2})}^{2} x})) + \log (\sqrt[3]{x^{2}})$

Pull terms out from under the radical.

$\frac{1}{2} \cdot (\log (x^{4}) - \log (x^{2} \sqrt{x})) + \log (\sqrt[3]{x^{2}})$

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

$\frac{1}{2} \cdot \log (\frac{x^{4}}{x^{2} \sqrt{x}}) + \log (\sqrt[3]{x^{2}})$

Cancel the common factor of $x^{4}$ and $x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

$\frac{1}{2} \cdot \log (\frac{x^{2} x^{2}}{x^{2} \sqrt{x}}) + \log (\sqrt[3]{x^{2}})$

Cancel the common factors.

Factor $x^{2}$ out of $x^{2} \sqrt{x}$ .

$\frac{1}{2} \cdot \log (\frac{x^{2} x^{2}}{x^{2} (\sqrt{x})}) + \log (\sqrt[3]{x^{2}})$

Cancel the common factor.

$\frac{1}{2} \cdot \log (\frac{x^{2} x^{2}}{x^{2} \sqrt{x}}) + \log (\sqrt[3]{x^{2}})$

Rewrite the expression.

$\frac{1}{2} \cdot \log (\frac{x^{2}}{\sqrt{x}}) + \log (\sqrt[3]{x^{2}})$

Multiply $\frac{x^{2}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{1}{2} \cdot \log (\frac{x^{2}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}) + \log (\sqrt[3]{x^{2}})$

Combine and simplify the denominator.

Multiply $\frac{x^{2}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{\sqrt{x} \sqrt{x}}) + \log (\sqrt[3]{x^{2}})$

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}) + \log (\sqrt[3]{x^{2}})$

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}) + \log (\sqrt[3]{x^{2}})$

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

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}) + \log (\sqrt[3]{x^{2}})$

Add $1$ and $1$ .

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{{\sqrt{x}}^{2}}) + \log (\sqrt[3]{x^{2}})$

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

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

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}) + \log (\sqrt[3]{x^{2}})$

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

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}) + \log (\sqrt[3]{x^{2}})$

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

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{x^{\frac{2}{2}}}) + \log (\sqrt[3]{x^{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{x^{\frac{2}{2}}}) + \log (\sqrt[3]{x^{2}})$

Rewrite the expression.

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{x^{1}}) + \log (\sqrt[3]{x^{2}})$

Simplify.

$\frac{1}{2} \cdot \log (\frac{x^{2} \sqrt{x}}{x}) + \log (\sqrt[3]{x^{2}})$

Cancel the common factor of $x^{2}$ and $x$ .

Factor $x$ out of $x^{2} \sqrt{x}$ .

$\frac{1}{2} \cdot \log (\frac{x (x \sqrt{x})}{x}) + \log (\sqrt[3]{x^{2}})$

Cancel the common factors.

Raise $x$ to the power of $1$ .

$\frac{1}{2} \cdot \log (\frac{x (x \sqrt{x})}{x^{1}}) + \log (\sqrt[3]{x^{2}})$

Factor $x$ out of $x^{1}$ .

$\frac{1}{2} \cdot \log (\frac{x (x \sqrt{x})}{x \cdot 1}) + \log (\sqrt[3]{x^{2}})$

Cancel the common factor.

$\frac{1}{2} \cdot \log (\frac{x (x \sqrt{x})}{x \cdot 1}) + \log (\sqrt[3]{x^{2}})$

Rewrite the expression.

$\frac{1}{2} \cdot \log (\frac{x \sqrt{x}}{1}) + \log (\sqrt[3]{x^{2}})$

Divide $x \sqrt{x}$ by $1$ .

$\frac{1}{2} \cdot \log (x \sqrt{x}) + \log (\sqrt[3]{x^{2}})$

Simplify $\frac{1}{2} \log (x \sqrt{x})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log ({(x \sqrt{x})}^{\frac{1}{2}}) + \log (\sqrt[3]{x^{2}})$

Apply the product rule to $x \sqrt{x}$ .

$\log (x^{\frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}}) + \log (\sqrt[3]{x^{2}})$

Step 2

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

$\log (x^{\frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}} \sqrt[3]{x^{2}})$

Step 3

Multiply $x^{\frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}} \sqrt[3]{x^{2}}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\log (x^{\frac{2}{3}} x^{\frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}})$

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

$\log (x^{\frac{2}{3} + \frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}})$

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log (x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}})$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log (x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}} {\sqrt{x}}^{\frac{1}{2}})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$\log (x^{\frac{2 \cdot 2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}} {\sqrt{x}}^{\frac{1}{2}})$

Multiply $3$ by $2$ .

$\log (x^{\frac{2 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3}} {\sqrt{x}}^{\frac{1}{2}})$

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

$\log (x^{\frac{2 \cdot 2}{6} + \frac{3}{2 \cdot 3}} {\sqrt{x}}^{\frac{1}{2}})$

Multiply $2$ by $3$ .

$\log (x^{\frac{2 \cdot 2}{6} + \frac{3}{6}} {\sqrt{x}}^{\frac{1}{2}})$

Combine the numerators over the common denominator.

$\log (x^{\frac{2 \cdot 2 + 3}{6}} {\sqrt{x}}^{\frac{1}{2}})$

Simplify the numerator.

Multiply $2$ by $2$ .

$\log (x^{\frac{4 + 3}{6}} {\sqrt{x}}^{\frac{1}{2}})$

Add $4$ and $3$ .

$\log (x^{\frac{7}{6}} {\sqrt{x}}^{\frac{1}{2}})$

Write as a Single Logarithm 1/2(4 log of x- log of square root of x^5)+ log of cube root of x^2

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