What is (4ln(x-6)+ln(x^2+6)-7ln(x+3))^{1/5} ?

The solution to (4ln(x-6)+ln(x^2+6)-7ln(x+3))^{1/5} is ln^{1/5}(((x-6)^4(x^2+6))/((x+3)^7))

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simplify (4ln(x-6)+ln(x^2+6)-7ln(x+3))^{1/5}

simplify

(4 ln (x - 6) + ln (x^{2} + 6) - 7 ln (x + 3))^{\frac{1}{5}}

ln^{\frac{1}{5}} (\frac{( x - 6 ) ^{4} ( x ^{2} + 6 )}{( x + 3 ) ^{7}})

Solution steps

(4 ln (x - 6) + ln (x^{2} + 6) - 7 ln (x + 3))^{\frac{1}{5}}

Apply log rule:

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

4 ln (x - 6) = ln ((x - 6)^{4})

= (ln ((x - 6)^{4}) + ln (x^{2} + 6) - 7 ln (x + 3))^{\frac{1}{5}}

Apply log rule:

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

ln ((x - 6)^{4}) + ln (x^{2} + 6) = ln ((x - 6)^{4} (x^{2} + 6))

= (ln ((x - 6)^{4} (x^{2} + 6)) - 7 ln (x + 3))^{\frac{1}{5}}

Apply log rule:

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

7 ln (x + 3) = ln ((x + 3)^{7})

= (ln ((x - 6)^{4} (x^{2} + 6)) - ln ((x + 3)^{7}))^{\frac{1}{5}}

Apply log rule:

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

ln ((x - 6)^{4} (x^{2} + 6)) - ln ((x + 3)^{7}) = ln (\frac{( x - 6 ) ^{4} ( x ^{2} + 6 )}{( x + 3 ) ^{7}})

= ln^{\frac{1}{5}} (\frac{( x - 6 ) ^{4} ( x ^{2} + 6 )}{( x + 3 ) ^{7}})

200 - 160 (2 ln (3) - 2 ln (2))

4 lo g_{2} (z) - \frac{1}{2} lo g_{2} (w) + 6 lo g_{2} (y)

2 lo g_{10} (5.6) + 4 lo g_{10} (3.5)

\frac{1}{2} ln (2 \cdot 9.6 + 1) + 2 ln (9.6 - 1)

- lo g_{10} (6.46 \cdot 1 0^{- 5})

What is (4ln(x-6)+ln(x^2+6)-7ln(x+3))^{1/5} ?
The solution to (4ln(x-6)+ln(x^2+6)-7ln(x+3))^{1/5} is ln^{1/5}(((x-6)^4(x^2+6))/((x+3)^7))