What is ln(3)+1/2 ln(x+2)-5ln(1+sqrt(x)) ?

The solution to ln(3)+1/2 ln(x+2)-5ln(1+sqrt(x)) is ln((3(x+2)^{1/2})/((1+sqrt(x))^5))

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simplify ln(3)+1/2 ln(x+2)-5ln(1+sqrt(x))

simplify

ln (3) + \frac{1}{2} ln (x + 2) - 5 ln (1 + x)

ln (\frac{3 ( x + 2 ) ^{\frac{1}{2}}}{( 1 + x ) ^{5}})

Solution steps

ln (3) + \frac{1}{2} ln (x + 2) - 5 ln (1 + x)

Apply log rule:

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

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

= ln (3) + ln ((x + 2)^{\frac{1}{2}}) - 5 ln (x + 1)

Apply log rule:

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

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

= ln (3 (x + 2)^{\frac{1}{2}}) - 5 ln (1 + x)

Apply log rule:

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

5 ln (x + 1) = ln ((x + 1)^{5})

= ln (3 (x + 2)^{\frac{1}{2}}) - ln ((x + 1)^{5})

Apply log rule:

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

ln (3 (x + 2)^{\frac{1}{2}}) - ln ((1 + x)^{5}) = ln (\frac{3 ( x + 2 ) ^{\frac{1}{2}}}{( x + 1 ) ^{5}})

= ln (\frac{3 ( x + 2 ) ^{\frac{1}{2}}}{( 1 + x ) ^{5}})

ln (x + 2) - ln (x^{2} + 2 x)

(2.5 \cdot ln (2) + ln (3)) \cdot 0.831

2000 (ln (140000) - ln (87500)) - 245

6 lo g_{4} (y - 5) - 5 lo g_{4} (y + 4)

6 lo g_{b} (w) - \frac{1}{5} lo g_{b} (z) + 5 lo g_{b} (x)

What is ln(3)+1/2 ln(x+2)-5ln(1+sqrt(x)) ?
The solution to ln(3)+1/2 ln(x+2)-5ln(1+sqrt(x)) is ln((3(x+2)^{1/2})/((1+sqrt(x))^5))