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

The solution to 3ln(x+2)+2ln(3x)-1/2 ln(y+1) is ln((9x^2(x+2)^3sqrt(y+1))/(y+1))

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

simplify

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

ln (\frac{9 x ^{2} ( x + 2 ) ^{3} y + 1}{y + 1})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

(x + 2)^{3} (3 x)^{2} : 9 x^{2} (x + 2)^{3}

= ln (9 x^{2} (x + 2)^{3}) - \frac{1}{2} ln (y + 1)

Apply log rule:

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

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

= ln (9 x^{2} (x + 2)^{3}) - ln ((y + 1)^{\frac{1}{2}})

Apply log rule:

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

ln (9 x^{2} (x + 2)^{3}) - ln ((y + 1)^{\frac{1}{2}}) = ln (\frac{9 x ^{2} ( x + 2 ) ^{3}}{( y + 1 ) ^{\frac{1}{2}}})

= ln (\frac{9 x ^{2} ( x + 2 ) ^{3}}{( y + 1 ) ^{\frac{1}{2}}})

Simplify

\frac{9 x ^{2} ( x + 2 ) ^{3}}{( y + 1 ) ^{\frac{1}{2}}} : \frac{9 x ^{2} ( x + 2 ) ^{3} y + 1}{y + 1}

= ln (\frac{9 x ^{2} ( x + 2 ) ^{3} y + 1}{y + 1})

\frac{( 1 )}{( 2 ) ( lo g _{5} ( x - 8 ) - lo g _{5} ( x ) )}

lo g_{2} (\frac{9}{8})

lo g_{5} (\frac{9 x ^{2} - y ^{2}}{2 ( 3 x + y )})

lo g_{10} (lo g_{10} (n) - lo g_{10} (49))

2 lo g_{a} (z + 1) + lo g_{a} (3 z + 2)

What is 3ln(x+2)+2ln(3x)-1/2 ln(y+1) ?
The solution to 3ln(x+2)+2ln(3x)-1/2 ln(y+1) is ln((9x^2(x+2)^3sqrt(y+1))/(y+1))