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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

Remove parentheses:

(a) = a

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

ln (\frac{10}{9.97})

6 lo g_{3} (w - 4) - 3 lo g_{3} (w + 3)

\frac{1}{4} ln (2) - \frac{1}{8} ln (\frac{4}{3}) + \frac{π}{24 3}

lo g_{5} (\frac{\frac{3 x + 1 ^{3}}{2 x - 1 ^{2}}}{x})

ln (\frac{7163}{7})

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