What is 3ln(2)+2ln(x)-1/2 ln(x+4) ?

The solution to 3ln(2)+2ln(x)-1/2 ln(x+4) is ln((8x^2sqrt(x+4))/(x+4))

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simplify 3ln(2)+2ln(x)-1/2 ln(x+4)

simplify

3 ln (2) + 2 ln (x) - \frac{1}{2} ln (x + 4)

ln (\frac{8 x ^{2} x + 4}{x + 4})

Solution steps

3 ln (2) + 2 ln (x) - \frac{1}{2} ln (x + 4)

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 + 4)

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 + 4)

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 + 4)

2^{3} = 8

= ln (8 x^{2}) - \frac{1}{2} ln (x + 4)

Apply log rule:

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

\frac{1}{2} ln (x + 4) = ln ((x + 4)^{\frac{1}{2}})

= ln (8 x^{2}) - ln ((x + 4)^{\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 + 4)^{\frac{1}{2}}) = ln (\frac{8 x ^{2}}{( x + 4 ) ^{\frac{1}{2}}})

= ln (\frac{8 x ^{2}}{( x + 4 ) ^{\frac{1}{2}}})

Simplify

\frac{8 x ^{2}}{( x + 4 ) ^{\frac{1}{2}}} : \frac{8 x ^{2} x + 4}{x + 4}

= ln (\frac{8 x ^{2} x + 4}{x + 4})

lo g_{8} (\frac{3 x ^{3}}{( 4 ) ( 3 )})

\frac{1}{4} lo g_{c} (x) + 4 lo g_{c} (y) - 3 lo g_{c} (x)

- lo g_{10} (5 \cdot 1 0^{- 4})

ln (\frac{2}{4})

4 lo g_{7} (x) - 3 lo g_{7} (y)

What is 3ln(2)+2ln(x)-1/2 ln(x+4) ?
The solution to 3ln(2)+2ln(x)-1/2 ln(x+4) is ln((8x^2sqrt(x+4))/(x+4))