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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

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

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

\frac{1}{2} ln (x) + 6 ln (y) - 7 ln (z)

- (ln (3 x))^{2} + 10 ln (3 x)

- \frac{2}{3} lo g_{5} (5 m^{2}) + \frac{1}{2} lo g_{5} (25 m^{2})

lo g_{4} (3 z) + lo g_{4} (x)

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