vereinfachen 1/3 ln(x)+10/12 ln(x+3)-1/6 ln(x-3)

Lösung

vereinfachen

\frac{1}{3} ln (x) + \frac{10}{12} ln (x + 3) - \frac{1}{6} ln (x - 3)

ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3})

Schritte zur Lösung

\frac{1}{3} ln (x) + \frac{10}{12} ln (x + 3) - \frac{1}{6} ln (x - 3)

Wende die log Regel an:

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

\frac{1}{3} ln (x) = ln (x^{\frac{1}{3}})

= ln (x^{\frac{1}{3}}) + \frac{10}{12} ln (x + 3) - \frac{1}{6} ln (x - 3)

Wende die log Regel an:

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

\frac{10}{12} ln (x + 3) = ln ((x + 3)^{\frac{10}{12}})

= ln (x^{\frac{1}{3}}) + ln ((x + 3)^{\frac{10}{12}}) - \frac{1}{6} ln (x - 3)

Wende die log Regel an:

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

ln (x^{\frac{1}{3}}) + ln ((x + 3)^{\frac{10}{12}}) = ln (x^{\frac{1}{3}} (x + 3)^{\frac{10}{12}})

= ln (x^{\frac{1}{3}} (x + 3)^{\frac{10}{12}}) - \frac{1}{6} ln (x - 3)

(x + 3)^{\frac{10}{12}} = (x + 3)^{\frac{5}{6}}

= ln (x^{\frac{1}{3}} (x + 3)^{\frac{5}{6}}) - \frac{1}{6} ln (x - 3)

Wende die log Regel an:

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

\frac{1}{6} ln (x - 3) = ln ((x - 3)^{\frac{1}{6}})

= ln (x^{\frac{1}{3}} (x + 3)^{\frac{5}{6}}) - ln ((x - 3)^{\frac{1}{6}})

Wende die log Regel an:

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

ln (x^{\frac{1}{3}} (x + 3)^{\frac{5}{6}}) - ln ((x - 3)^{\frac{1}{6}}) = ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{6}}})

= ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{6}}})

Vereinfache

\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{6}}} : \frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3}

= ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3})

s im pl i f y \frac{1}{3} (5 ln (x + 6) - ln (x) - ln (x^{2} - 25))

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

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

s im pl i f y ln (\frac{17}{23})

s im pl i f y - lo g_{10} (5.8 \cdot 1 0^{- 4})