# xlog4 m+1/2 log 4 n-log4 p

xlog4 m+1/2 log 4 n-log4 p

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$x\mathrm{log}\left(4m\right)+\frac{1}{2}\mathrm{log}\left(4n\right)-\mathrm{log}\left(4p\right)$

Simplify each term.

Simplify $\frac{1}{2}\mathrm{log}\left(4n\right)$ by moving $\frac{1}{2}$ inside the logarithm.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({\left(4n\right)}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Apply the product rule to $4n$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({4}^{\frac{1}{2}}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Rewrite $4$ as ${2}^{2}$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({\left({2}^{2}\right)}^{\frac{1}{2}}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

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

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({2}^{2\left(\frac{1}{2}\right)}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Cancel the common factor of $2$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left({2}^{1}{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Evaluate the exponent.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(2{n}^{\frac{1}{2}}\right)-\mathrm{log}\left(4p\right)$

Use the quotient property of logarithms, ${\mathrm{log}}_{b}\left(x\right)-{\mathrm{log}}_{b}\left(y\right)={\mathrm{log}}_{b}\left(\frac{x}{y}\right)$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(\frac{2{n}^{\frac{1}{2}}}{4p}\right)$

Simplify each term.

Factor $2$ out of $2{n}^{\frac{1}{2}}$.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(\frac{2\left({n}^{\frac{1}{2}}\right)}{4p}\right)$

Cancel the common factors.

$x\mathrm{log}\left(4m\right)+\mathrm{log}\left(\frac{{n}^{\frac{1}{2}}}{2p}\right)$