What is ln(x)-3ln(y)+1/2 ln(z) ?

The solution to ln(x)-3ln(y)+1/2 ln(z) is ln((xz^{1/2})/(y^3))

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simplify ln(x)-3ln(y)+1/2 ln(z)

simplify

ln (x) - 3 ln (y) + \frac{1}{2} ln (z)

ln (\frac{x z ^{\frac{1}{2}}}{y ^{3}})

Solution steps

ln (x) - 3 ln (y) + \frac{1}{2} ln (z)

Apply log rule:

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

3 ln (y) = ln (y^{3})

= ln (x) - ln (y^{3}) + \frac{1}{2} ln (z)

Apply log rule:

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

ln (x) - ln (y^{3}) = ln (\frac{x}{y ^{3}})

= ln (\frac{x}{y ^{3}}) + \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 (\frac{x}{y ^{3}}) + ln (z^{\frac{1}{2}})

Apply log rule:

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

ln (\frac{x}{y ^{3}}) + ln (z^{\frac{1}{2}}) = ln (\frac{x}{y ^{3}} z^{\frac{1}{2}})

= ln (\frac{x}{y ^{3}} z^{\frac{1}{2}})

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= ln (\frac{x z ^{\frac{1}{2}}}{y ^{3}})

lo g_{b} (\frac{0.34}{3 0.83})

ln (\frac{1}{6.5})

ln (\frac{0.02}{0.01})

ln (\frac{3000000}{1256526})

lo g_{5} (\frac{3 x ^{3} ( 1 - 2 x ) ^{6}}{2 x ^{4} ( x ^{2} - y ^{2} )})

What is ln(x)-3ln(y)+1/2 ln(z) ?
The solution to ln(x)-3ln(y)+1/2 ln(z) is ln((xz^{1/2})/(y^3))