vereinfachen 3[ln(x-2)+2ln(x+1)-ln(x+2)-5ln(x-1)]

Lösung

vereinfachen

3 [ln (x - 2) + 2 ln (x + 1) - ln (x + 2) - 5 ln (x - 1)]

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

Schritte zur Lösung

3 [ln (x - 2) + 2 ln (x + 1) - ln (x + 2) - 5 ln (x - 1)]

Wende die log Regel an:

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

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

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

Wende die log Regel an:

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

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

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

Wende die log Regel an:

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

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

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

Wende die log Regel an:

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

5 ln (x - 1) = ln ((x - 1)^{5})

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

Wende die log Regel an:

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

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

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

Wende Bruchregel an:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

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

Entferne die Klammern:

(a) = a

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

s im pl i f y lo g_{10} (100 \cdot x)

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

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

s im pl i f y 2 lo g_{a} (z + 1) + lo g_{a} (3 z + 6)

s im pl i f y ln (\frac{x}{y}) + ln (\frac{x}{z})