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

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

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

simplify

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

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

Solution steps

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

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

Apply log rule:

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

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

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

Apply log rule:

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

3 ln (x + 1) = ln ((x + 1)^{3})

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

Apply log rule:

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

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

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

\frac{1}{2} ln (x) + 8 ln (y) - 9 ln (z)

ln (\frac{9}{49})

ln (\frac{3}{2}) - ln (\frac{1}{2})

ln (\frac{1.2}{0.6})

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