What is 1/2 ln(x)+1/2 ln(y)-1/2 ln(z) ?

The solution to 1/2 ln(x)+1/2 ln(y)-1/2 ln(z) is ln((x^{1/2}y^{1/2}sqrt(z))/z)

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simplify 1/2 ln(x)+1/2 ln(y)-1/2 ln(z)

simplify

\frac{1}{2} ln (x) + \frac{1}{2} ln (y) - \frac{1}{2} ln (z)

ln (\frac{x ^{\frac{1}{2}} y ^{\frac{1}{2}} z}{z})

Solution steps

\frac{1}{2} ln (x) + \frac{1}{2} ln (y) - \frac{1}{2} ln (z)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} ln (x) = ln (x^{\frac{1}{2}})

= ln (x^{\frac{1}{2}}) + \frac{1}{2} ln (y) - \frac{1}{2} ln (z)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} ln (y) = ln (y^{\frac{1}{2}})

= ln (x^{\frac{1}{2}}) + ln (y^{\frac{1}{2}}) - \frac{1}{2} ln (z)

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (x^{\frac{1}{2}}) + ln (y^{\frac{1}{2}}) = ln (x^{\frac{1}{2}} y^{\frac{1}{2}})

= ln (x^{\frac{1}{2}} y^{\frac{1}{2}}) - \frac{1}{2} ln (z)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} ln (z) = ln (z^{\frac{1}{2}})

= ln (x^{\frac{1}{2}} y^{\frac{1}{2}}) - ln (z^{\frac{1}{2}})

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (x^{\frac{1}{2}} y^{\frac{1}{2}}) - ln (z^{\frac{1}{2}}) = ln (\frac{x ^{\frac{1}{2}} y ^{\frac{1}{2}}}{z ^{\frac{1}{2}}})

= ln (\frac{x ^{\frac{1}{2}} y ^{\frac{1}{2}}}{z ^{\frac{1}{2}}})

Simplify

\frac{x ^{\frac{1}{2}} y ^{\frac{1}{2}}}{z ^{\frac{1}{2}}} : \frac{x ^{\frac{1}{2}} y ^{\frac{1}{2}} z}{z}

= ln (\frac{x ^{\frac{1}{2}} y ^{\frac{1}{2}} z}{z})

lo g_{5} (2 x (8 x + 7))

ln (ln (4)) - ln (ln (2))

lo g_{10} (y) + 11 lo g_{10} (z)

lo g_{10} (\frac{100 x ^{2}}{27})

What is 1/2 ln(x)+1/2 ln(y)-1/2 ln(z) ?
The solution to 1/2 ln(x)+1/2 ln(y)-1/2 ln(z) is ln((x^{1/2}y^{1/2}sqrt(z))/z)