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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

ln (x^{2} - 4) - ln (x - 2)

64 lo g_{4} (3) + 36 lo g_{\frac{1}{6}} (\frac{1}{3})

- 10 ln (\frac{85.7 - 70}{98.6 - 70})

ln (x) - 2 [ln (x - 5) + ln (x + 5)]

lo g_{10} (\frac{x y}{z t})

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