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Beliebt Algebra >

vereinfachen ln(96/7)+1/2 (ln(2/3 x))+1/4 (ln(15/8))

Lösung

vereinfachen

ln (\frac{96}{7}) + \frac{1}{2} (ln (\frac{2}{3} x)) + \frac{1}{4} (ln (\frac{15}{8}))

Lösung

\frac{1}{2} ln (x) - ln (7) - \frac{1}{4} ln (2) - \frac{1}{2} ln (3) + ln (96) + \frac{1}{4} ln (15)

Schritte zur Lösung

ln (\frac{96}{7}) + \frac{1}{2} (ln (\frac{2}{3} x)) + \frac{1}{4} (ln (\frac{15}{8}))

Wende die log Regel an:

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

ln (\frac{96}{7}) = ln (96) - ln (7)

= ln (96) - ln (7) + \frac{1}{2} ln (\frac{2}{3} x) + \frac{1}{4} ln (\frac{15}{8})

Wende die log Regel an:

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

ln (\frac{15}{8}) = ln (15) - ln (8)

= ln (96) - ln (7) + \frac{1}{2} ln (\frac{2}{3} x) + \frac{1}{4} (ln (15) - ln (8))

Wende die log Regel an:

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

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

= ln (96) - ln (7) + \frac{1}{2} ln (\frac{2}{3}) + \frac{1}{2} ln (x) + \frac{1}{4} (ln (15) - ln (8))

Wende die log Regel an:

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

ln (\frac{2}{3}) = ln (2) - ln (3)

= ln (96) - ln (7) + \frac{1}{2} (ln (2) - ln (3)) + \frac{1}{2} ln (x) + \frac{1}{4} (ln (15) - ln (8))

ln (8) = 3 ln (2)

= ln (96) - ln (7) + \frac{1}{2} (ln (2) - ln (3)) + \frac{1}{2} ln (x) + \frac{1}{4} (ln (15) - 3 ln (2))

Multipliziere aus

\frac{1}{2} (ln (2) - ln (3)) : \frac{1}{2} ln (2) - \frac{1}{2} ln (3)

= ln (96) - ln (7) + \frac{1}{2} ln (2) - \frac{1}{2} ln (3) + \frac{1}{2} ln (x) + \frac{1}{4} (ln (15) - 3 ln (2))

Multipliziere aus

\frac{1}{4} (ln (15) - 3 ln (2)) : \frac{1}{4} ln (15) - \frac{3}{4} ln (2)

= ln (96) - ln (7) + \frac{1}{2} ln (2) - \frac{1}{2} ln (3) + \frac{1}{2} ln (x) + \frac{1}{4} ln (15) - \frac{3}{4} ln (2)

Vereinfache

ln (96) - ln (7) + \frac{1}{2} ln (2) - \frac{1}{2} ln (3) + \frac{1}{2} ln (x) + \frac{1}{4} ln (15) - \frac{3}{4} ln (2) : \frac{1}{2} ln (x) - ln (7) - \frac{1}{4} ln (2) - \frac{1}{2} ln (3) + ln (96) + \frac{1}{4} ln (15)

= \frac{1}{2} ln (x) - ln (7) - \frac{1}{4} ln (2) - \frac{1}{2} ln (3) + ln (96) + \frac{1}{4} ln (15)

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