What is 1/2 ln(2x+5)-1/3 ln(x+8)+ln(C) ?

The solution to 1/2 ln(2x+5)-1/3 ln(x+8)+ln(C) is ln((C(2x+5)^{1/2}(x+8)^{2/3})/(x+8))

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simplify 1/2 ln(2x+5)-1/3 ln(x+8)+ln(C)

simplify

\frac{1}{2} ln (2 x + 5) - \frac{1}{3} ln (x + 8) + ln (C)

ln (\frac{C ( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8})

Solution steps

\frac{1}{2} ln (2 x + 5) - \frac{1}{3} ln (x + 8) + 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 + 8) + ln (C)

Apply log rule:

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

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

= ln ((2 x + 5)^{\frac{1}{2}}) - ln ((x + 8)^{\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 + 8)^{\frac{1}{3}}) = ln (\frac{( 2 x + 5 ) ^{\frac{1}{2}}}{( x + 8 ) ^{\frac{1}{3}}})

= ln (\frac{( 2 x + 5 ) ^{\frac{1}{2}}}{( x + 8 ) ^{\frac{1}{3}}}) + ln (C)

Simplify

\frac{( 2 x + 5 ) ^{\frac{1}{2}}}{( x + 8 ) ^{\frac{1}{3}}} : \frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8}

= ln (\frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8}) + 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 + 8 ) ^{\frac{2}{3}}}{x + 8}) + ln (C) = ln (C \frac{( x + 8 ) ^{\frac{2}{3}} ( 2 x + 5 ) ^{\frac{1}{2}}}{x + 8})

= ln (C \frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8})

Simplify

\frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8} C : \frac{C ( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8}

= ln (\frac{C ( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8})

- lo g_{10} (7.83 \cdot 1 0^{- 8})

ln (8 \cdot 12)

ln (\frac{( 75 + x )}{75})

\frac{1}{ln ( 2 )} \cdot ln (\frac{4}{3})

(0.287) ln (\frac{500}{200}) - (1.05) ln (\frac{680}{980})

What is 1/2 ln(2x+5)-1/3 ln(x+8)+ln(C) ?
The solution to 1/2 ln(2x+5)-1/3 ln(x+8)+ln(C) is ln((C(2x+5)^{1/2}(x+8)^{2/3})/(x+8))