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

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

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

simplify

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

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

Solution steps

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

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}}) + 3 ln (y) - 2 ln (x)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

lo g_{4} (u) + 6 lo g_{4} (v) + 3 lo g_{4} (w)

\frac{1}{2} lo g_{10} (x) + 6 lo g_{10} (y)

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

- (\frac{3}{14}) lo g_{2} (\frac{3}{14}) - (\frac{2}{14}) lo g_{2} (\frac{2}{14})

ln (\frac{28500}{19250})

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