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

The solution to 3ln(2)+5ln(x)-1/2 ln(x+3) is ln((8x^5sqrt(x+3))/(x+3))

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

simplify

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

ln (\frac{8 x ^{5} x + 3}{x + 3})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

5 ln (x) = ln (x^{5})

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

Apply log rule:

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

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

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

2^{3} = 8

= ln (8 x^{5}) - \frac{1}{2} ln (x + 3)

Apply log rule:

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

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

= ln (8 x^{5}) - ln ((x + 3)^{\frac{1}{2}})

Apply log rule:

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

ln (8 x^{5}) - ln ((x + 3)^{\frac{1}{2}}) = ln (\frac{8 x ^{5}}{( x + 3 ) ^{\frac{1}{2}}})

= ln (\frac{8 x ^{5}}{( x + 3 ) ^{\frac{1}{2}}})

Simplify

\frac{8 x ^{5}}{( x + 3 ) ^{\frac{1}{2}}} : \frac{8 x ^{5} x + 3}{x + 3}

= ln (\frac{8 x ^{5} x + 3}{x + 3})

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

lo g_{2} (x) - 5 lo g_{2} (y)

lo g_{5} (x + 12) + lo g_{5} (x - 12)

lo g_{2} (\frac{7 r ^{5}}{z ^{7}})

ln (\frac{1}{N !})

What is 3ln(2)+5ln(x)-1/2 ln(x+3) ?
The solution to 3ln(2)+5ln(x)-1/2 ln(x+3) is ln((8x^5sqrt(x+3))/(x+3))