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vereinfachen 4/5 ln(x)-2ln(y^4)+5ln(w)-7ln(z)

Lösung

vereinfachen

\frac{4}{5} ln (x) - 2 ln (y^{4}) + 5 ln (w) - 7 ln (z)

ln (\frac{w ^{5} x ^{\frac{4}{5}}}{y ^{8} z ^{7}})

Schritte zur Lösung

\frac{4}{5} ln (x) - 2 ln (y^{4}) + 5 ln (w) - 7 ln (z)

Wende die log Regel an:

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

\frac{4}{5} ln (x) = ln (x^{\frac{4}{5}})

= ln (x^{\frac{4}{5}}) - 2 ln (y^{4}) + 5 ln (w) - 7 ln (z)

Wende die log Regel an:

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

2 ln (y^{4}) = ln ((y^{4})^{2})

= ln (x^{\frac{4}{5}}) - ln ((y^{4})^{2}) + 5 ln (w) - 7 ln (z)

Wende die log Regel an:

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

ln (x^{\frac{4}{5}}) - ln ((y^{4})^{2}) = ln (\frac{x ^{\frac{4}{5}}}{( y ^{4} ) ^{2}})

= ln (\frac{x ^{\frac{4}{5}}}{( y ^{4} ) ^{2}}) + 5 ln (w) - 7 ln (z)

(y^{4})^{2} = y^{8}

= ln (\frac{x ^{\frac{4}{5}}}{y ^{8}}) + 5 ln (w) - 7 ln (z)

Wende die log Regel an:

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

5 ln (w) = ln (w^{5})

= ln (\frac{x ^{\frac{4}{5}}}{y ^{8}}) + ln (w^{5}) - 7 ln (z)

Wende die log Regel an:

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

ln (\frac{x ^{\frac{4}{5}}}{y ^{8}}) + ln (w^{5}) = ln (\frac{x ^{\frac{4}{5}}}{y ^{8}} w^{5})

= ln (\frac{x ^{\frac{4}{5}}}{y ^{8}} w^{5}) - 7 ln (z)

Multipliziere Brüche:

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

= ln (\frac{w ^{5} x ^{\frac{4}{5}}}{y ^{8}}) - 7 ln (z)

Wende die log Regel an:

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

7 ln (z) = ln (z^{7})

= ln (\frac{w ^{5} x ^{\frac{4}{5}}}{y ^{8}}) - ln (z^{7})

Wende die log Regel an:

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

ln (\frac{w ^{5} x ^{\frac{4}{5}}}{y ^{8}}) - ln (z^{7}) = ln \frac{\frac{w ^{5} x ^{\frac{4}{5}}}{y ^{8}}}{z ^{7}}

= ln \frac{\frac{w ^{5} x ^{\frac{4}{5}}}{y ^{8}}}{z ^{7}}

Wende Bruchregel an:

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

= ln (\frac{w ^{5} x ^{\frac{4}{5}}}{y ^{8} z ^{7}})

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

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

s im pl i f y lo g_{10} (\frac{4 ( x + 8 ) ^{5}}{3 x ^{2}})

s im pl i f y lo g_{5} (\frac{625}{y})

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