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lehrkraefte:ks:miniex:ex05 [2019/02/16 21:00] Simon Knaus |
lehrkraefte:ks:miniex:ex05 [2019/02/16 21:28] (current) Simon Knaus [Aufgabe 3] |
} | } |
</JS> | </JS> |
=== Freitag 2. November 2018 === | ==== Aufgabe 1 ==== |
| Berechne die folgenden Audrücke von Hand: |
| - $\log_3(27)$ |
| - $\log_{27}(81)$ |
| - $\log_{16}(2)$ |
| - $\log_{4}(\sqrt{2})$ |
| |
| <hidden Lösungen> |
| Zur Erinnerung: Es ist immer $\log_{a}(b)=c\Leftrightarrow a^c=b$. In Worten: $\log_a(b)$ beantwortet die Frage <<$a$ hoch wie viel gibt $b$?>>. |
| |
| - $3^{3}=27 \Rightarrow \log_3{27}=3$ |
| - $27^{\frac{4}{3}}=81\Rightarrow \log_{27}{81}=\frac{4}{3}$ |
| - $16^{\frac14}=2\Rightarrow\log_{16}(2)=\frac14$ |
| - $4^{\frac{1}{4}}=2^{\frac{1}{2}}\Rightarrow\log_{4}(\sqrt{2})=\frac{1}{4}$ |
| </hidden> |
| ==== Aufgabe 2 ==== |
| |
Lösen Sie die Gleichung von Hand auf: | Lösen Sie die Gleichung von Hand auf: |
<HTML> | <HTML> |
<div id="solexpgleichung"></div> | <div id="solexpgleichung"></div> |
| </HTML> |
| </hidden> |
| |
| <JS> |
| function generate(jQuery, idex, idsol, ex, sep="<br>", sep2="<br>") { |
| var drei=function(n) { |
| var one = Math.floor(Math.random()*(n-2)); |
| var two = Math.floor(Math.random()*(n-2)); |
| if (one>two) { |
| var h = one; |
| one = two; |
| two = h; |
| } |
| var three = Math.floor(Math.random()*(n-2)); |
| if (two>=one) { |
| two++; |
| } |
| if (three>=one) { |
| three++; |
| } |
| if (three>=two) { |
| three++; |
| } |
| return [one,two,three]; |
| }; |
| |
| var selec=drei(ex.length); |
| //console.log(selec); |
| |
| for (var i=0; i<3; i++) { |
| //console.log(selec[i]); |
| jQuery(idex).append((i+1)+". "+ex[selec[i]][0]+sep); |
| jQuery(idsol).append((i+1)+". "+ex[selec[i]][1]+sep2); |
| } |
| |
| } |
| </JS> |
| |
| |
| |
| |
| |
| |
| |
| |
| ==== Aufgabe 3 ==== |
| Berechne it Hilfe eines geeigneten Basiswechsels: |
| <JS>jQuery(function() {generate(jQuery, "#exobasiswechsel","#solbasiswechsel", |
| [["$\\log_{25}(\\frac{1}{625})$", "$\\log_{25}(\\frac{1}{625})$=$\\frac{\\log_{5}\\left(5^{-4}\\right)}{\\log_{5}\\left(5^{2}\\right)}=\\frac{-4}{2}=-\\frac{2}{1}$"], ["$\\log_{\\frac{1}{32}}(256)$", "$\\log_{\\frac{1}{32}}(256)$=$\\frac{\\log_{2}\\left(2^{8}\\right)}{\\log_{2}\\left(2^{-5}\\right)}=\\frac{8}{-5}=-\\frac{8}{5}$"], ["$\\log_{\\frac{1}{8}}(32)$", "$\\log_{\\frac{1}{8}}(32)$=$\\frac{\\log_{2}\\left(2^{5}\\right)}{\\log_{2}\\left(2^{-3}\\right)}=\\frac{5}{-3}=-\\frac{5}{3}$"], ["$\\log_{\\frac{1}{512}}(16)$", "$\\log_{\\frac{1}{512}}(16)$=$\\frac{\\log_{2}\\left(2^{4}\\right)}{\\log_{2}\\left(2^{-9}\\right)}=\\frac{4}{-9}=-\\frac{4}{9}$"], ["$\\log_{64}(\\frac{1}{128})$", "$\\log_{64}(\\frac{1}{128})$=$\\frac{\\log_{2}\\left(2^{-7}\\right)}{\\log_{2}\\left(2^{6}\\right)}=\\frac{-7}{6}=-\\frac{7}{6}$"], ["$\\log_{\\frac{1}{8}}(512)$", "$\\log_{\\frac{1}{8}}(512)$=$\\frac{\\log_{2}\\left(2^{9}\\right)}{\\log_{2}\\left(2^{-3}\\right)}=\\frac{9}{-3}=-\\frac{3}{1}$"], ["$\\log_{\\frac{1}{8}}(128)$", "$\\log_{\\frac{1}{8}}(128)$=$\\frac{\\log_{2}\\left(2^{7}\\right)}{\\log_{2}\\left(2^{-3}\\right)}=\\frac{7}{-3}=-\\frac{7}{3}$"], ["$\\log_{\\frac{1}{512}}(64)$", "$\\log_{\\frac{1}{512}}(64)$=$\\frac{\\log_{2}\\left(2^{6}\\right)}{\\log_{2}\\left(2^{-9}\\right)}=\\frac{6}{-9}=-\\frac{2}{3}$"], ["$\\log_{64}(\\frac{1}{512})$", "$\\log_{64}(\\frac{1}{512})$=$\\frac{\\log_{2}\\left(2^{-9}\\right)}{\\log_{2}\\left(2^{6}\\right)}=\\frac{-9}{6}=-\\frac{3}{2}$"], ["$\\log_{\\frac{1}{32}}(8)$", "$\\log_{\\frac{1}{32}}(8)$=$\\frac{\\log_{2}\\left(2^{3}\\right)}{\\log_{2}\\left(2^{-5}\\right)}=\\frac{3}{-5}=-\\frac{3}{5}$"]], |
| " <br> ");}); |
| </JS> |
| <HTML> |
| <div id="exobasiswechsel"></div> |
| |
| </HTML> |
| <hidden Lösungen> |
| <HTML> |
| <div id="solbasiswechsel"></div> |
| </HTML> |
| </hidden> |
| ==== Auftrag 4 ==== |
| Berechnen Sie von Hand (Repetieren Sie dazu 2er Potenzen bis $2^{10}$, 3er Potenzen bis $3^4$, damit 4er bis $4^5$ und 5er Potenzen bis $5^4$). |
| <JS>jQuery(function() {generate(jQuery, "#exologpot","#sollogpot", |
| [["$\\log_{4}\\left(\\frac{1}{1024}\\right)+\\log_{3}\\left(27\\right)+\\log_{4}\\left(\\frac{1}{64}\\right)$", "$-5+3+-3=-5$"], ["$\\log_{5}\\left(\\frac{1}{25}\\right)+\\log_{2}\\left(64\\right)+\\log_{2}\\left(\\frac{1}{1024}\\right)$", "$-2+6+-10=-6$"], ["$\\log_{2}\\left(\\frac{1}{64}\\right)+\\log_{2}\\left(512\\right)+\\log_{5}\\left(\\frac{1}{625}\\right)$", "$-6+9+-4=-1$"], ["$\\log_{4}\\left(256\\right)+\\log_{2}\\left(\\frac{1}{128}\\right)+\\log_{4}\\left(64\\right)$", "$4+-7+3=0$"], ["$\\log_{2}\\left(\\frac{1}{512}\\right)+\\log_{3}\\left(81\\right)+\\log_{2}\\left(\\frac{1}{256}\\right)$", "$-9+4+-8=-13$"], ["$\\log_{3}\\left(\\frac{1}{27}\\right)+\\log_{3}\\left(\\frac{1}{81}\\right)+\\log_{2}\\left(32\\right)$", "$-3+-4+5=-2$"], ["$\\log_{5}\\left(25\\right)+\\log_{5}\\left(625\\right)+\\log_{2}\\left(128\\right)$", "$2+4+7=13$"], ["$\\log_{2}\\left(\\frac{1}{32}\\right)+\\log_{4}\\left(\\frac{1}{256}\\right)+\\log_{2}\\left(256\\right)$", "$-5+-4+8=-1$"], ["$\\log_{5}\\left(\\frac{1}{125}\\right)+\\log_{5}\\left(125\\right)+\\log_{4}\\left(1024\\right)$", "$-3+3+5=5$"]], |
| " <br> ");}); |
| </JS> |
| <HTML> |
| <div id="exologpot"></div> |
| |
| </HTML> |
| <hidden Lösungen> |
| <HTML> |
| <div id="sollogpot"></div> |
| </HTML> |
| </hidden> |
| ==== Aufgabe 5 ==== |
| Schreibe als Summe von Vielfachen von $\log_a(x)$ und $\log_a(y)$. Verwende dabei die Logarithmus Gesetze. |
| <JS>jQuery(function() {generate(jQuery, "#exologlaws","#solloglaws", |
| [["$\\log_a\\left(\\left(x^{4}y^{-3}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{3}y^{4}\\right)^{4}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{4}y^{-3}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{3}y^{4}\\right)^{4}\\right)=\\log_a\\left(\\left(x^{4}y^{-3}\\right)^{5} \\cdot \\left(x^{3}y^{4}\\right)^{4}\\right) = \\\\\\log_a\\left(x^{20}y^{-15} \\cdot x^{12}y^{16}\\right) =\\log_a\\left(x^{32} \\cdot y^{1} \\right) = \\\\\\log_a\\left(x^{32}\\right) + \\log_a\\left(y^{1} \\right) = 32 \\log_a(x) +1\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{3}y^{4}\\right)^{-3}\\right)+\\log_a\\left(\\left(x^{2}y^{-3}\\right)^{-5}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{3}y^{4}\\right)^{-3}\\right)+\\log_a\\left(\\left(x^{2}y^{-3}\\right)^{-5}\\right)=\\log_a\\left(\\left(x^{3}y^{4}\\right)^{-3} \\cdot \\left(x^{2}y^{-3}\\right)^{-5}\\right) = \\\\\\log_a\\left(x^{-9}y^{-12} \\cdot x^{-10}y^{15}\\right) =\\log_a\\left(x^{-19} \\cdot y^{3} \\right) = \\\\\\log_a\\left(x^{-19}\\right) + \\log_a\\left(y^{3} \\right) = -19 \\log_a(x) +3\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{5}y^{-5}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{-2}y^{-4}\\right)^{-5}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{5}y^{-5}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{-2}y^{-4}\\right)^{-5}\\right)=\\log_a\\left(\\left(x^{5}y^{-5}\\right)^{5} \\cdot \\left(x^{-2}y^{-4}\\right)^{-5}\\right) = \\\\\\log_a\\left(x^{25}y^{-25} \\cdot x^{10}y^{20}\\right) =\\log_a\\left(x^{35} \\cdot y^{-5} \\right) = \\\\\\log_a\\left(x^{35}\\right) + \\log_a\\left(y^{-5} \\right) = 35 \\log_a(x) -5\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{3}y^{-3}\\right)^{-5}\\right)+\\log_a\\left(\\left(x^{4}y^{-4}\\right)^{2}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{3}y^{-3}\\right)^{-5}\\right)+\\log_a\\left(\\left(x^{4}y^{-4}\\right)^{2}\\right)=\\log_a\\left(\\left(x^{3}y^{-3}\\right)^{-5} \\cdot \\left(x^{4}y^{-4}\\right)^{2}\\right) = \\\\\\log_a\\left(x^{-15}y^{15} \\cdot x^{8}y^{-8}\\right) =\\log_a\\left(x^{-7} \\cdot y^{7} \\right) = \\\\\\log_a\\left(x^{-7}\\right) + \\log_a\\left(y^{7} \\right) = -7 \\log_a(x) +7\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{5}y^{4}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{3}y^{-4}\\right)^{2}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{5}y^{4}\\right)^{5}\\right)+\\log_a\\left(\\left(x^{3}y^{-4}\\right)^{2}\\right)=\\log_a\\left(\\left(x^{5}y^{4}\\right)^{5} \\cdot \\left(x^{3}y^{-4}\\right)^{2}\\right) = \\\\\\log_a\\left(x^{25}y^{20} \\cdot x^{6}y^{-8}\\right) =\\log_a\\left(x^{31} \\cdot y^{12} \\right) = \\\\\\log_a\\left(x^{31}\\right) + \\log_a\\left(y^{12} \\right) = 31 \\log_a(x) +12\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{3}y^{-3}\\right)^{-5}\\right)+\\log_a\\left(\\left(x^{2}y^{3}\\right)^{4}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{3}y^{-3}\\right)^{-5}\\right)+\\log_a\\left(\\left(x^{2}y^{3}\\right)^{4}\\right)=\\log_a\\left(\\left(x^{3}y^{-3}\\right)^{-5} \\cdot \\left(x^{2}y^{3}\\right)^{4}\\right) = \\\\\\log_a\\left(x^{-15}y^{15} \\cdot x^{8}y^{12}\\right) =\\log_a\\left(x^{-7} \\cdot y^{27} \\right) = \\\\\\log_a\\left(x^{-7}\\right) + \\log_a\\left(y^{27} \\right) = -7 \\log_a(x) +27\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{5}y^{-5}\\right)^{2}\\right)+\\log_a\\left(\\left(x^{4}y^{5}\\right)^{5}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{5}y^{-5}\\right)^{2}\\right)+\\log_a\\left(\\left(x^{4}y^{5}\\right)^{5}\\right)=\\log_a\\left(\\left(x^{5}y^{-5}\\right)^{2} \\cdot \\left(x^{4}y^{5}\\right)^{5}\\right) = \\\\\\log_a\\left(x^{10}y^{-10} \\cdot x^{20}y^{25}\\right) =\\log_a\\left(x^{30} \\cdot y^{15} \\right) = \\\\\\log_a\\left(x^{30}\\right) + \\log_a\\left(y^{15} \\right) = 30 \\log_a(x) +15\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{-3}y^{-2}\\right)^{-2}\\right)+\\log_a\\left(\\left(x^{-3}y^{4}\\right)^{5}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{-3}y^{-2}\\right)^{-2}\\right)+\\log_a\\left(\\left(x^{-3}y^{4}\\right)^{5}\\right)=\\log_a\\left(\\left(x^{-3}y^{-2}\\right)^{-2} \\cdot \\left(x^{-3}y^{4}\\right)^{5}\\right) = \\\\\\log_a\\left(x^{6}y^{4} \\cdot x^{-15}y^{20}\\right) =\\log_a\\left(x^{-9} \\cdot y^{24} \\right) = \\\\\\log_a\\left(x^{-9}\\right) + \\log_a\\left(y^{24} \\right) = -9 \\log_a(x) +24\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{-4}y^{-5}\\right)^{-2}\\right)+\\log_a\\left(\\left(x^{3}y^{4}\\right)^{3}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{-4}y^{-5}\\right)^{-2}\\right)+\\log_a\\left(\\left(x^{3}y^{4}\\right)^{3}\\right)=\\log_a\\left(\\left(x^{-4}y^{-5}\\right)^{-2} \\cdot \\left(x^{3}y^{4}\\right)^{3}\\right) = \\\\\\log_a\\left(x^{8}y^{10} \\cdot x^{9}y^{12}\\right) =\\log_a\\left(x^{17} \\cdot y^{22} \\right) = \\\\\\log_a\\left(x^{17}\\right) + \\log_a\\left(y^{22} \\right) = 17 \\log_a(x) +22\\log_a(y)\\end{multline*}$"], ["$\\log_a\\left(\\left(x^{2}y^{-3}\\right)^{3}\\right)+\\log_a\\left(\\left(x^{-3}y^{5}\\right)^{-5}\\right)$", "$\\begin{multline*}\\log_a\\left(\\left(x^{2}y^{-3}\\right)^{3}\\right)+\\log_a\\left(\\left(x^{-3}y^{5}\\right)^{-5}\\right)=\\log_a\\left(\\left(x^{2}y^{-3}\\right)^{3} \\cdot \\left(x^{-3}y^{5}\\right)^{-5}\\right) = \\\\\\log_a\\left(x^{6}y^{-9} \\cdot x^{15}y^{-25}\\right) =\\log_a\\left(x^{21} \\cdot y^{-34} \\right) = \\\\\\log_a\\left(x^{21}\\right) + \\log_a\\left(y^{-34} \\right) = 21 \\log_a(x) -34\\log_a(y)\\end{multline*}$"]], |
| " <br> ");}); |
| </JS> |
| <HTML> |
| <div id="exologlaws"></div> |
| |
| </HTML> |
| <hidden Lösungen> |
| <HTML> |
| <div id="solloglaws"></div> |
</HTML> | </HTML> |
</hidden> | </hidden> |
| |