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

The solution to 1/3 [2ln(x+3)+ln(x)-ln(x^2-1)] is 1/3 ln((x(x+3)^2)/(x^2-1))

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Remove parentheses:

(a) = a

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

lo g_{10} (3) + lo g_{10} (2)

lo g_{10} (12) + \frac{1}{2} lo g_{10} (7) - lo g_{10} (2)

lo g_{10} (3) + lo g_{10} (4)

lo g_{10} (\frac{6 ( x + 4 ) ^{3}}{x ^{5}})

ln (5 e^{6})

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