vereinfachen ln((3x^4\sqrt[5]{(x-1)^3})/((3x+1)^2))

Lösung

vereinfachen

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

ln (3) + 4 ln (x) + ln (5 (x - 1)^{3}) - 2 ln (3 x + 1)

Schritte zur Lösung

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

Wende die log Regel an:

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

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

= ln (3 x^{4} 5 (x - 1)^{3}) - ln ((3 x + 1)^{2})

Wende die log Regel an:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= ln (3 x^{4} 5 (x - 1)^{3}) - 2 ln (3 x + 1)

Wende die log Regel an:

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

ln (3 x^{4} 5 (x - 1)^{3}) = ln (3) + ln (x^{4}) + ln (5 (x - 1)^{3})

= ln (3) + ln (x^{4}) + ln (5 (x - 1)^{3}) - 2 ln (3 x + 1)

Wende die log Regel an:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= ln (3) + 4 ln (x) + ln (5 (x - 1)^{3}) - 2 ln (3 x + 1)

s im pl i f y \frac{1}{3} ln (x) + \frac{10}{12} ln (x + 3) - \frac{1}{6} ln (x - 3)

s im pl i f y \frac{1}{3} (5 ln (x + 6) - ln (x) - ln (x^{2} - 25))

s im pl i f y - \frac{1}{2} ln 1 - (\frac{y}{x})^{2} + ln \frac{y}{x} + 1 - ln \frac{y}{x} - 1

s im pl i f y 2 lo g_{2} (x) + lo g_{2} (x + 5)

s im pl i f y ln (\frac{17}{23})