Algebra Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Simplify $\frac{3}{4} \log (x^{4})$ by moving $\frac{3}{4}$ inside the logarithm.

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

Step 2.2

Multiply the exponents in ${(x^{4})}^{\frac{3}{4}}$ .

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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Step 4.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$\log (\sqrt{x} x)$