What is 1/2 [4ln(x+1)+ln(x)-ln(x^4-3)] ?

The solution to 1/2 [4ln(x+1)+ln(x)-ln(x^4-3)] is 1/2 ln((x(x+1)^4)/(x^4-3))

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simplify 1/2 [4ln(x+1)+ln(x)-ln(x^4-3)]

simplify

\frac{1}{2} [4 ln (x + 1) + ln (x) - ln (x^{4} - 3)]

\frac{1}{2} ln (\frac{x ( x + 1 ) ^{4}}{x ^{4} - 3})

Solution steps

\frac{1}{2} [4 ln (x + 1) + ln (x) - ln (x^{4} - 3)]

Apply log rule:

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

4 ln (x + 1) = ln ((x + 1)^{4})

= \frac{1}{2} [ln ((x + 1)^{4}) + ln (x) - ln (x^{4} - 3)]

Apply log rule:

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

ln ((x + 1)^{4}) + ln (x) = ln (x (x + 1)^{4})

= \frac{1}{2} [ln ((x + 1)^{4} x) - ln (x^{4} - 3)]

Apply log rule:

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

ln (x (x + 1)^{4}) - ln (x^{4} - 3) = ln (\frac{x ( x + 1 ) ^{4}}{x ^{4} - 3})

= \frac{1}{2} [ln (\frac{( x + 1 ) ^{4} x}{x ^{4} - 3})]

Remove parentheses:

(a) = a

= \frac{1}{2} ln (\frac{( x + 1 ) ^{4} x}{x ^{4} - 3})

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

\frac{1}{2} ln (x) - ln (9)

lo g_{x} (\frac{1}{81})

3 (ln (ln (4)) - ln (ln (2)))

lo g_{2} (R) + lo g_{2} (S) - 2 lo g_{2} (T)

What is 1/2 [4ln(x+1)+ln(x)-ln(x^4-3)] ?
The solution to 1/2 [4ln(x+1)+ln(x)-ln(x^4-3)] is 1/2 ln((x(x+1)^4)/(x^4-3))