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vereinfachen 7/4 ln(x)-5/4 ln(y)-1/4 ln(z)+1/4 ln(x)

Lösung

vereinfachen

\frac{7}{4} ln (x) - \frac{5}{4} ln (y) - \frac{1}{4} ln (z) + \frac{1}{4} ln (x)

ln (\frac{x ^{\frac{5}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{z y ^{\frac{5}{4}}})

Schritte zur Lösung

\frac{7}{4} ln (x) - \frac{5}{4} ln (y) - \frac{1}{4} ln (z) + \frac{1}{4} ln (x)

Wende die log Regel an:

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

\frac{7}{4} ln (x) = ln (x^{\frac{7}{4}})

= ln (x^{\frac{7}{4}}) - \frac{5}{4} ln (y) - \frac{1}{4} ln (z) + \frac{1}{4} ln (x)

Wende die log Regel an:

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

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

= ln (x^{\frac{7}{4}}) - ln (y^{\frac{5}{4}}) - \frac{1}{4} ln (z) + \frac{1}{4} ln (x)

Wende die log Regel an:

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

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

= ln (\frac{x ^{\frac{7}{4}}}{y ^{\frac{5}{4}}}) - \frac{1}{4} ln (z) + \frac{1}{4} ln (x)

Fasse gleiche Potenzen zusammen:

\frac{x ^{\frac{a}{m}}}{y ^{\frac{b}{m}}} = m \frac{x ^{a}}{y ^{b}}

= ln (4 \frac{x ^{7}}{y ^{5}}) - \frac{1}{4} ln (z) + \frac{1}{4} ln (x)

Wende die log Regel an:

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

\frac{1}{4} ln (z) = ln (z^{\frac{1}{4}})

= ln (4 \frac{x ^{7}}{y ^{5}}) - ln (z^{\frac{1}{4}}) + \frac{1}{4} ln (x)

Wende die log Regel an:

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

ln (4 \frac{x ^{7}}{y ^{5}}) - ln (z^{\frac{1}{4}}) = ln \frac{4 \frac{x ^{7}}{y ^{5}}}{z ^{\frac{1}{4}}}

= ln \frac{4 \frac{x ^{7}}{y ^{5}}}{z ^{\frac{1}{4}}} + \frac{1}{4} ln (x)

Vereinfache

\frac{4 \frac{x ^{7}}{y ^{5}}}{z ^{\frac{1}{4}}} : \frac{x y ^{\frac{3}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{y ^{2} z}

= ln (\frac{x y ^{\frac{3}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{y ^{2} z}) + \frac{1}{4} ln (x)

Wende die log Regel an:

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

\frac{1}{4} ln (x) = ln (x^{\frac{1}{4}})

= ln (\frac{x y ^{\frac{3}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{y ^{2} z}) + ln (x^{\frac{1}{4}})

Wende die log Regel an:

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

ln (\frac{x y ^{\frac{3}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{y ^{2} z}) + ln (x^{\frac{1}{4}}) = ln (\frac{x y ^{\frac{3}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{y ^{2} z} x^{\frac{1}{4}})

= ln (\frac{x y ^{\frac{3}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{y ^{2} z} x^{\frac{1}{4}})

Vereinfache

\frac{x y ^{\frac{3}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{y ^{2} z} x^{\frac{1}{4}} : \frac{x ^{\frac{5}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{z y ^{\frac{5}{4}}}

= ln (\frac{x ^{\frac{5}{4}} z ^{\frac{3}{4}} 4 x ^{3}}{z y ^{\frac{5}{4}}})

s im pl i f y ln (\frac{2}{e})

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

e x p an d lo g_{10} (\frac{x ^{2}}{y ^{4}})

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

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