What is 1/3 [5ln(x+6)+ln(x)-ln(x^2-6)] ?

The solution to 1/3 [5ln(x+6)+ln(x)-ln(x^2-6)] is 1/3 ln((x(x+6)^5)/(x^2-6))

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simplify 1/3 [5ln(x+6)+ln(x)-ln(x^2-6)]

simplify

\frac{1}{3} [5 ln (x + 6) + ln (x) - ln (x^{2} - 6)]

\frac{1}{3} ln (\frac{x ( x + 6 ) ^{5}}{x ^{2} - 6})

Solution steps

\frac{1}{3} [5 ln (x + 6) + ln (x) - ln (x^{2} - 6)]

Apply log rule:

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

5 ln (x + 6) = ln ((x + 6)^{5})

= \frac{1}{3} [ln ((x + 6)^{5}) + ln (x) - ln (x^{2} - 6)]

Apply log rule:

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

ln ((x + 6)^{5}) + ln (x) = ln (x (x + 6)^{5})

= \frac{1}{3} [ln ((x + 6)^{5} x) - ln (x^{2} - 6)]

Apply log rule:

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

ln (x (x + 6)^{5}) - ln (x^{2} - 6) = ln (\frac{x ( x + 6 ) ^{5}}{x ^{2} - 6})

= \frac{1}{3} [ln (\frac{( x + 6 ) ^{5} x}{x ^{2} - 6})]

Remove parentheses:

(a) = a

= \frac{1}{3} ln (\frac{( x + 6 ) ^{5} x}{x ^{2} - 6})

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What is 1/3 [5ln(x+6)+ln(x)-ln(x^2-6)] ?
The solution to 1/3 [5ln(x+6)+ln(x)-ln(x^2-6)] is 1/3 ln((x(x+6)^5)/(x^2-6))