Write as a Single Logarithm log of square root of x- log of x^2+3/4 log of x^4

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

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

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

Simplify each term.

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

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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