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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

5 ln (y^{2} + 1) = ln ((y^{2} + 1)^{5})

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

Apply log rule:

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

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

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

\frac{1}{2} ln (3) - ln (7)

ln (\frac{x}{e ^{9 x}})

lo g_{10} (\frac{21}{x ^{2}})

ln ((5 p^{3} q^{4})^{2})

\frac{1}{2} lo g_{3} (\frac{( 27 b ^{2} )}{c})

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