What is 3ln(2)+2ln(x)-1/2 ln(x+8) ?

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

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simplify 3ln(2)+2ln(x)-1/2 ln(x+8)

simplify

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

ln (\frac{8 x ^{2} x + 8}{x + 8})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

2^{3} = 8

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

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