What is 1/2 ln(x)-5ln(y)+4ln(z) ?

The solution to 1/2 ln(x)-5ln(y)+4ln(z) is ln((z^4x^{1/2})/(y^5))

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simplify 1/2 ln(x)-5ln(y)+4ln(z)

simplify

\frac{1}{2} ln (x) - 5 ln (y) + 4 ln (z)

ln (\frac{z ^{4} x ^{\frac{1}{2}}}{y ^{5}})

Solution steps

\frac{1}{2} ln (x) - 5 ln (y) + 4 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) + 4 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}) + 4 ln (z)

Apply log rule:

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

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

= ln (\frac{x ^{\frac{1}{2}}}{y ^{5}}) + 4 ln (z)

Apply log rule:

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

4 ln (z) = ln (z^{4})

= ln (\frac{x ^{\frac{1}{2}}}{y ^{5}}) + ln (z^{4})

Apply log rule:

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

ln (\frac{x ^{\frac{1}{2}}}{y ^{5}}) + ln (z^{4}) = ln (\frac{x ^{\frac{1}{2}}}{y ^{5}} z^{4})

= ln (\frac{x ^{\frac{1}{2}}}{y ^{5}} z^{4})

Multiply fractions:

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

= ln (\frac{z ^{4} x ^{\frac{1}{2}}}{y ^{5}})

2 ln (a) - ln (3 b) + 2 ln (2)

lo g_{10} (3 x) - 1

lo g_{b} (19 x^{5} y)

lo g_{10} (\frac{y ^{6}}{z})

What is 1/2 ln(x)-5ln(y)+4ln(z) ?
The solution to 1/2 ln(x)-5ln(y)+4ln(z) is ln((z^4x^{1/2})/(y^5))