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Beliebt Algebra >

vereinfachen ln(y/x-4)+6(1/5 ln(y/x-1)-1/5 ln(y/x+4))+ln(x)

Lösung

vereinfachen

ln (\frac{y}{x} - 4) + 6 (\frac{1}{5} ln (\frac{y}{x} - 1) - \frac{1}{5} ln (\frac{y}{x} + 4)) + ln (x)

Lösung

ln ((\frac{y - x}{y + 4 x})^{\frac{6}{5}} (y - 4 x))

Schritte zur Lösung

ln (\frac{y}{x} - 4) + 6 (\frac{1}{5} ln (\frac{y}{x} - 1) - \frac{1}{5} ln (\frac{y}{x} + 4)) + ln (x)

Wende die log Regel an:

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

ln (\frac{y}{x} - 4) + ln (x) = ln (x (\frac{y}{x} - 4))

= ln (x (\frac{y}{x} - 4)) + 6 (\frac{1}{5} ln (\frac{y}{x} - 1) - \frac{1}{5} ln (\frac{y}{x} + 4))

Vereinfache

(\frac{y}{x} - 4) x : y - 4 x

= ln (y - 4 x) + 6 (\frac{1}{5} ln (\frac{y}{x} - 1) - \frac{1}{5} ln (\frac{y}{x} + 4))

Wende die log Regel an:

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

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

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

Wende die log Regel an:

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

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

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

Wende die log Regel an:

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

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

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

Vereinfache

\frac{( \frac{y}{x} - 1 ) ^{\frac{1}{5}}}{( \frac{y}{x} + 4 ) ^{\frac{1}{5}}} : 5 \frac{y - x}{y + 4 x}

= ln (y - 4 x) + 6 ln (5 \frac{y - x}{y + 4 x})

Wende die log Regel an:

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

6 ln (5 \frac{y - x}{y + 4 x}) = ln ((5 \frac{y - x}{y + 4 x})^{6})

= ln (y - 4 x) + ln ((5 \frac{y - x}{y + 4 x})^{6})

Wende die log Regel an:

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

ln (y - 4 x) + ln ((5 \frac{y - x}{y + 4 x})^{6}) = ln ((5 \frac{y - x}{y + 4 x})^{6} (y - 4 x))

= ln ((5 \frac{y - x}{y + 4 x})^{6} (y - 4 x))

(5 \frac{y - x}{y + 4 x})^{6} = (\frac{y - x}{y + 4 x})^{\frac{6}{5}}

= ln ((\frac{y - x}{y + 4 x})^{\frac{6}{5}} (y - 4 x))

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