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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

4 ln (x) = ln (x^{4})

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

Apply log rule:

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

ln (2^{3}) + ln (x^{4}) = ln (2^{3} x^{4})

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

2^{3} = 8

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

lo g_{10} (7) + lo g_{10} (18) - lo g_{10} (9)

lo g_{4} (x) - 5 lo g_{4} (y) + 2 lo g_{4} (z)

ln (5) + 6 ln (x) - 5 ln (y)

ln (\frac{x}{a} \cdot e^{\frac{- x ^{2}}{( 2 a )}})

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