What is ln(x)-1/2 (3ln(x)+5ln(y)) ?

The solution to ln(x)-1/2 (3ln(x)+5ln(y)) is ln((sqrt(x))/(xy^{5/2)})

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simplify ln(x)-1/2 (3ln(x)+5ln(y))

simplify

ln (x) - \frac{1}{2} (3 ln (x) + 5 ln (y))

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

Solution steps

ln (x) - \frac{1}{2} (3 ln (x) + 5 ln (y))

Apply log rule:

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

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

= ln (x) - \frac{1}{2} (ln (x^{3}) + 5 ln (y))

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 (x^{3}) + ln (y^{5}))

Apply log rule:

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

ln (x^{3}) + ln (y^{5}) = ln (x^{3} y^{5})

= ln (x) - \frac{1}{2} ln (x^{3} y^{5})

Apply log rule:

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

\frac{1}{2} ln (x^{3} y^{5}) = ln ((x^{3} y^{5})^{\frac{1}{2}})

= ln (x) - ln ((x^{3} y^{5})^{\frac{1}{2}})

Apply log rule:

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

ln (x) - ln ((x^{3} y^{5})^{\frac{1}{2}}) = ln (\frac{x}{( x ^{3} y ^{5} ) ^{\frac{1}{2}}})

= ln (\frac{x}{( x ^{3} y ^{5} ) ^{\frac{1}{2}}})

Simplify

\frac{x}{( x ^{3} y ^{5} ) ^{\frac{1}{2}}} : \frac{x}{x y ^{\frac{5}{2}}}

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

3 ln (4) - 4 ln (3)

ln (x) - [ln (x + 1) + ln (x - 1)]

lo g_{b} (x) + lo g_{b} (y) - lo g_{b} (u)

\frac{1}{2} lo g_{b} (x) + 5 lo g_{b} (y) - 4 lo g_{b} (x)

lo g_{3} (x y)

What is ln(x)-1/2 (3ln(x)+5ln(y)) ?
The solution to ln(x)-1/2 (3ln(x)+5ln(y)) is ln((sqrt(x))/(xy^{5/2)})