vereinfachen ln|z+4|+6/5 ln|z-1|-6/5 ln|z+4|+ln|x|

Lösung

vereinfachen

ln ∣ z + 4 ∣ + \frac{6}{5} ln ∣ z - 1 ∣ - \frac{6}{5} ln ∣ z + 4 ∣ + ln ∣ x ∣

ln (\frac{∣ x ∣ ∣ z - 1 ∣ ^{\frac{6}{5}} ∣ z + 4 ∣ ^{\frac{4}{5}}}{∣ z + 4 ∣})

Schritte zur Lösung

ln ∣ z + 4 ∣ + \frac{6}{5} ln ∣ z - 1 ∣ - \frac{6}{5} ln ∣ z + 4 ∣ + ln ∣ x ∣

Wende die log Regel an:

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

\frac{6}{5} ln ∣ z - 1 ∣ = ln (∣ z - 1 ∣^{\frac{6}{5}})

= ln ∣ z + 4 ∣ + ln (∣ z - 1 ∣^{\frac{6}{5}}) - \frac{6}{5} ln ∣ z + 4 ∣ + ln ∣ x ∣

Wende die log Regel an:

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

ln ∣ z + 4 ∣ + ln (∣ z - 1 ∣^{\frac{6}{5}}) = ln (∣ z + 4 ∣ ∣ z - 1 ∣^{\frac{6}{5}})

= ln (∣ z + 4 ∣ ∣ z - 1 ∣^{\frac{6}{5}}) - \frac{6}{5} ln ∣ z + 4 ∣ + ln ∣ x ∣

Wende die log Regel an:

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

\frac{6}{5} ln ∣ z + 4 ∣ = ln (∣ z + 4 ∣^{\frac{6}{5}})

= ln (∣ z + 4 ∣ ∣ z - 1 ∣^{\frac{6}{5}}) - ln (∣ z + 4 ∣^{\frac{6}{5}}) + ln ∣ x ∣

Wende die log Regel an:

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

ln (∣ z + 4 ∣ ∣ z - 1 ∣^{\frac{6}{5}}) - ln (∣ z + 4 ∣^{\frac{6}{5}}) = ln (\frac{∣ z + 4 ∣ ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣ ^{\frac{6}{5}}})

= ln (\frac{∣ z + 4 ∣ ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣ ^{\frac{6}{5}}}) + ln ∣ x ∣

Vereinfache

\frac{∣ z + 4 ∣ ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣ ^{\frac{6}{5}}} : \frac{∣ z + 4 ∣ ^{\frac{4}{5}} ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣}

= ln (\frac{∣ z + 4 ∣ ^{\frac{4}{5}} ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣}) + ln ∣ x ∣

Wende die log Regel an:

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

ln (\frac{∣ z + 4 ∣ ^{\frac{4}{5}} ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣}) + ln ∣ x ∣ = ln (\frac{∣ z + 4 ∣ ^{\frac{4}{5}} ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣} ∣ x ∣)

= ln (\frac{∣ z + 4 ∣ ^{\frac{4}{5}} ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣} ∣ x ∣)

Vereinfache

\frac{∣ z + 4 ∣ ^{\frac{4}{5}} ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣} ∣ x ∣ : \frac{∣ x ∣ ∣ z - 1 ∣ ^{\frac{6}{5}} ∣ z + 4 ∣ ^{\frac{4}{5}}}{∣ z + 4 ∣}

= ln (\frac{∣ x ∣ ∣ z - 1 ∣ ^{\frac{6}{5}} ∣ z + 4 ∣ ^{\frac{4}{5}}}{∣ z + 4 ∣})

s im pl i f y \frac{1}{2} ln ∣ 2 (10.3) + 1 ∣ + 2 ln ∣ (10.3) - 1 ∣

s im pl i f y \frac{1}{4} ln (\frac{3 e ^{\frac{3}{2} x}}{3 e ^{\frac{3}{2} x} - 2})

s im pl i f y ln (\frac{2.3}{2.6})

s im pl i f y - 0.3 lo g_{2} (0.3) - 0.7 lo g_{2} (0.7)

s im pl i f y ln (Y \cdot e^{- a (i)})