What is 1/2 ln(x)+5ln(y)-2ln(z) ?

The solution to 1/2 ln(x)+5ln(y)-2ln(z) is ln((x^{1/2}y^5)/(z^2))

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simplify 1/2 ln(x)+5ln(y)-2ln(z)

simplify

\frac{1}{2} ln (x) + 5 ln (y) - 2 ln (z)

ln (\frac{x ^{\frac{1}{2}} y ^{5}}{z ^{2}})

Solution steps

\frac{1}{2} ln (x) + 5 ln (y) - 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}}) + 5 ln (y) - 2 ln (z)

Apply log rule:

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

5 ln (y) = ln (y^{5})

= ln (x^{\frac{1}{2}}) + ln (y^{5}) - 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^{5}) = ln (y^{5} x^{\frac{1}{2}})

= ln (x^{\frac{1}{2}} y^{5}) - 2 ln (z)

Apply log rule:

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

2 ln (z) = ln (z^{2})

= ln (y^{5} x^{\frac{1}{2}}) - ln (z^{2})

Apply log rule:

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

ln (y^{5} x^{\frac{1}{2}}) - ln (z^{2}) = ln (\frac{y ^{5} x ^{\frac{1}{2}}}{z ^{2}})

= ln (\frac{x ^{\frac{1}{2}} y ^{5}}{z ^{2}})

\frac{1}{2} lo g_{b} (x) + 2 lo g_{b} (y) - 3 lo g_{b} (x)

lo g_{10} (a b^{3})

lo g_{2} (\frac{y}{8})

ln (\frac{3 x ^{2}}{( x + 1 ) ^{10}})

What is 1/2 ln(x)+5ln(y)-2ln(z) ?
The solution to 1/2 ln(x)+5ln(y)-2ln(z) is ln((x^{1/2}y^5)/(z^2))