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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{3 x ^{\frac{1}{2}} y}{z ^{\frac{1}{3}}} : \frac{3 y x ^{\frac{1}{2}} z ^{\frac{2}{3}}}{z}

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

lo g_{10} (2) + lo g_{10} (9)

6 lo g_{10} (a) + lo g_{10} (b)

10 lo g_{10} (x) + \frac{2}{3} lo g_{10} (64)

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

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

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