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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (i (2 + 6))

ln (6) + ln (\frac{e ^{(2.46169)}}{3})

ln (\frac{3 ^{\frac{3}{8}}}{3})

ln (10 x^{2}) - ln (2 x^{2})

ln (\frac{8}{3.1 4 ^{2}} \cdot \frac{( 190 - 45 )}{65 - 45})

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