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

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

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simplify 1/2 ln(x+3)+1/3 ln(x+2)

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

lo g_{3} (\frac{3 x ^{5}}{y})

- ln (\frac{1}{4})

lo g_{10} (2 x^{4}) + lo g_{10} (3 x^{5})

ln (y) + ln (z)

lo g_{10} (\frac{y}{4 y + 8})

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