vereinfachen ln((x^2(1+x^2)^2)/((1-x^2)^2))

Lösung

vereinfachen

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

2 ln (x) + 2 ln (1 + x^{2}) - 2 ln (1 - x^{2})

Schritte zur Lösung

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

Wende die log Regel an:

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

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

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

Wende die log Regel an:

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

ln ((1 - x^{2})^{2}) = 2 ln (1 - x^{2})

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

Wende die log Regel an:

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

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

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

Wende die log Regel an:

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

ln (x^{2}) = 2 ln (x)

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

Wende die log Regel an:

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

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

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

s im pl i f y lo g_{6} (9) - lo g_{6} (4)

e x p an d lo g_{5} (x^{2} y^{4} 5 m^{3} p)

s im pl i f y lo g_{d} (x^{4} y^{7} z)

s im pl i f y 2 (lo g_{2} (5) + 2 lo g_{2} (3)) - lo g_{2} (15)

s im pl i f y lo g_{2} (\frac{4}{3 x ^{2}})