What is ln(9)+1/2 ln(x+4)-2ln(1+sqrt(x)) ?

The solution to ln(9)+1/2 ln(x+4)-2ln(1+sqrt(x)) is ln((9(x+4)^{1/2})/((1+sqrt(x))^2))

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simplify ln(9)+1/2 ln(x+4)-2ln(1+sqrt(x))

simplify

ln (9) + \frac{1}{2} ln (x + 4) - 2 ln (1 + x)

ln (\frac{9 ( x + 4 ) ^{\frac{1}{2}}}{( 1 + x ) ^{2}})

Solution steps

ln (9) + \frac{1}{2} ln (x + 4) - 2 ln (1 + x)

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

Apply log rule:

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

ln (9) + ln ((x + 4)^{\frac{1}{2}}) = ln (9 (x + 4)^{\frac{1}{2}})

= ln (9 (x + 4)^{\frac{1}{2}}) - 2 ln (1 + x)

Apply log rule:

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

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

= ln (9 (x + 4)^{\frac{1}{2}}) - ln ((x + 1)^{2})

Apply log rule:

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

ln (9 (x + 4)^{\frac{1}{2}}) - ln ((1 + x)^{2}) = ln (\frac{9 ( x + 4 ) ^{\frac{1}{2}}}{( x + 1 ) ^{2}})

= ln (\frac{9 ( x + 4 ) ^{\frac{1}{2}}}{( 1 + x ) ^{2}})

ln^{3} (2 x)

ln (\frac{58.5}{69800})

3 \cdot ln (5)

\frac{1}{4} \cdot ln (\frac{240}{75})

\frac{7}{3} ln (2)

What is ln(9)+1/2 ln(x+4)-2ln(1+sqrt(x)) ?
The solution to ln(9)+1/2 ln(x+4)-2ln(1+sqrt(x)) is ln((9(x+4)^{1/2})/((1+sqrt(x))^2))