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Beliebt Algebra >

vereinfachen 3/8ln(8/3)+10/16ln(16/10)

Lösung

vereinfachen

\frac{3}{8} \cdot ln (\frac{8}{3}) + \frac{10}{16} \cdot ln (\frac{16}{10})

Lösung

\frac{1}{8} ln (\frac{16777216}{84375})

+1

Dezimale

0.66156 \dots

Schritte zur Lösung

\frac{3}{8} ln (\frac{8}{3}) + \frac{10}{16} ln (\frac{16}{10})

Wende die log Regel an:

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

\frac{3}{8} ln (\frac{8}{3}) = ln ((\frac{8}{3})^{\frac{3}{8}})

= ln ((\frac{8}{3})^{\frac{3}{8}}) + \frac{10}{16} ln (\frac{16}{10})

Wende die log Regel an:

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

\frac{10}{16} ln (\frac{16}{10}) = ln ((\frac{16}{10})^{\frac{10}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}}) + ln ((\frac{16}{10})^{\frac{10}{16}})

Wende die log Regel an:

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

ln ((\frac{8}{3})^{\frac{3}{8}}) + ln ((\frac{16}{10})^{\frac{10}{16}}) = ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{10})^{\frac{10}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{10})^{\frac{10}{16}})

(\frac{16}{10})^{\frac{10}{16}} = (\frac{8}{5})^{\frac{5}{8}}

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{8}{5})^{\frac{5}{8}})

Wende Exponentenregel an:

a^{m} b^{m} = (ab)^{m}

(\frac{8}{3})^{\frac{3}{8}} (\frac{8}{5})^{\frac{5}{8}} = 8 (\frac{8}{5})^{5} (\frac{8}{3})^{3}

= ln 8 (\frac{8}{3})^{3} (\frac{8}{5})^{5}

Schreibe um

= ln ((\frac{8}{3})^{3} (\frac{8}{5})^{5})^{\frac{1}{8}}

Wende log Regel an

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

angenommen

x \geq 0

= \frac{1}{8} ln ((\frac{8}{5})^{5} (\frac{8}{3})^{3})

(\frac{8}{3})^{3} = \frac{8 ^{3}}{3 ^{3}}

= \frac{1}{8} ln ((\frac{8}{5})^{5} \frac{8 ^{3}}{3 ^{3}})

(\frac{8}{5})^{5} = \frac{8 ^{5}}{5 ^{5}}

= \frac{1}{8} ln (\frac{8 ^{3}}{3 ^{3}} \cdot \frac{8 ^{5}}{5 ^{5}})

Multipliziere

\frac{8 ^{3}}{3 ^{3}} \cdot \frac{8 ^{5}}{5 ^{5}} : \frac{16777216}{84375}

= \frac{1}{8} ln (\frac{16777216}{84375})

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