What is ln((x+3)^3)-2ln(sqrt(x))-2ln(x^2+3x) ?

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

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simplify ln((x+3)^3)-2ln(sqrt(x))-2ln(x^2+3x)

simplify

ln ((x + 3)^{3}) - 2 ln (x) - 2 ln (x^{2} + 3 x)

ln (\frac{x + 3}{x ^{3}})

Solution steps

ln ((x + 3)^{3}) - 2 ln (x) - 2 ln (x^{2} + 3 x)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

(x)^{2} = x

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{\frac{( x + 3 ) ^{3}}{x}}{( x ^{2} + 3 x ) ^{2}} : \frac{x + 3}{x ^{3}}

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

lo g_{10} (7 x^{4})

7 lo g_{a} (m) - \frac{1}{3} lo g_{a} (r)

ln (\frac{29}{22.6})

4 \cdot lo g_{10} (43488 3 \cdot 1 0^{- 2}) - 4

- \frac{2}{3} lo g_{3} (3 m^{2}) + \frac{1}{2} lo g_{3} (9 m^{2})

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