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1 -Logarithmus und Exponentialgleichungen
1 +BPE 4.5 Logarithmus und Exponentialgleichungen
Übergeordnete Seite
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1 -Main.WebHome
1 +Eingangsklasse.WebHome
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1 -XWiki.holger
1 +XWiki.martinrathgeb
Inhalt
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1 -{{box cssClass="floatinginfobox" title="**Contents**"}}
2 -{{toc start=2 depth=2 /}}
3 -{{/box}}
1 +{{seiteninhalt/}}
4 4  
5 -Die Schülerinnen und Schüler deuten den Logarithmus einer Zahl als Lösung einer Exponentialgleichung. Exponentialgleichungen lösen sie algebraisch und begründen die Auswahl der jeweiligen Lösungsstrategie. Die berechneten Lösungen interpretieren die Schülerinnen und Schüler grafisch als Nullstellen einer Funktion beziehungsweise als Schnittstellen zweier Funktionen.
3 +[[Kompetenzen.K5]] Ich kann Logarithmus nutzen, um eine Exponentialgleichung zu lösen
4 +[[Kompetenzen.K5]] Ich kann eine geeignete Strategie wählen, um eine gegebene Exponentialgleichung zu lösen
5 +[[Kompetenzen.K1]] Ich kann die Wahl einer Lösungsstrategie für eine Exponentialgleichung begründen
6 +[[Kompetenzen.K5]] Ich kann Exponentialgleichungen algebraisch lösen
7 +[[Kompetenzen.K4]] [[Kompetenzen.K6]] Ich kann die Lösungen einer Exponentialgleichung als Nullstelle interpretieren
8 +[[Kompetenzen.K4]] [[Kompetenzen.K6]] Ich kann die Lösungen einer Exponentialgleichung als Schnittstelle zweier Funktionen interpretieren
6 6  
10 +Aufgaben:
11 +– Logarithmus: graphisches Ermitteln vs. Operator
12 +Lösen von Exponentialgleichungen:
13 +– Vokabelheft für Umkehroperationen
14 +– Umkehrung der Rechenoperationen (Logarithmieren!) zzgl. Grundrechenarten
15 +– Faktorisierung durch Ausklammern und Satz vom Nullprodukt zzgl. Grundrechenarten
16 +– Substitution (abc-Formel, pq-Formel, Typ I) zzgl. Grundrechenarten
17 +- Näherungslösungen
18 +
19 +Gleichungen:
20 +x+y = e --> y = e - x
21 +x*y = e --> y = e / x
22 +e^y = x --> y = ln(x)
23 +
24 +{{aufgabe id="Exponentialgleichungen lösen (Fehlvorstellungen)" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="5"}}
25 +(% class="abc" %)
26 +1. (((Beurteile folgende Aussagen:
27 +1) Die Gleichung {{formula}} 5^x = 2 {{/formula}} kann ich nach x auflösen, indem ich durch 5 dividiere. Ich erhalte damit die Lösung {{formula}} x = \frac{2}{5} {{/formula}}.
28 +2) Die Gleichung {{formula}} 5^x = 2 {{/formula}} kann ich nach x auflösen, indem ich die 5-te Wurzel verwende. Ich erhalte damit die Lösung {{formula}} x = \sqrt[5]{2} {{/formula}}.
29 +3) Um die Gleichung {{formula}} 5^x = 2 {{/formula}} nach x aufzulösen, benötige ich eine neue Methode bzw. eine neue Operation.
30 +)))
31 +1. Umkehraufgabe: Gib für jede in a) falsche Methode eine passende Gleichung an.
32 +{{/aufgabe}}
33 +
34 +{{aufgabe id="Gleichungsformen instantiieren" afb="I" kompetenzen="K2,K5" quelle="Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
35 +Nenne möglichst viele (wahre) Gleichungen der folgenden Formen, wobei {{formula}} a, b, c \in \{2; 3; 4; \ldots; 16\} {{/formula}} gelten soll:
36 +{{formula}} c = a^b\:; \qquad c = \sqrt[a]{b}\:; \qquad c = \log_a(b)\:. {{/formula}}
37 +{{/aufgabe}}
38 +
39 +{{aufgabe id="Darstellungen zuordnen" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
40 +Ordne zu!
41 +(% class="abc" %)
42 +1. vier Gleichungen
43 +1. zwei Tabellen
44 +1. zwei Graphen
45 +{{/aufgabe}}
46 +
47 +{{aufgabe id="Logarithmen auswerten" afb="II" kompetenzen="K4,K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="10"}}
48 +Ordne (ohne WTR) die Terme ihren Werten gemäß den Kästchen über dem Zahlenstrahl zu. Trage dafür die jeweiligen Buchstaben in die Kästchen ein.
49 +
50 +[[image:Logarithmus.svg||width="600px"]]
51 +
52 +(% class="abc" %)
53 +1. {{formula}} \log_{10}(10) {{/formula}}
54 +1. {{formula}} \log_{100}(10) {{/formula}}
55 +1. {{formula}} \log_{11}(10) {{/formula}}
56 +1. {{formula}} \log_{10}(1000) {{/formula}}
57 +1. {{formula}} \log_{10}(5) {{/formula}}
58 +1. {{formula}} \log_{11}(1000) {{/formula}}
59 +1. {{formula}} \log_{10}(1) {{/formula}}
60 +1. {{formula}} \log_{100}(10) {{/formula}}
61 +1. {{formula}} \log_{10}(10) {{/formula}}
62 +{{/aufgabe}}
63 +
64 +{{aufgabe id="Exponentialgleichungen lösen (graphisch vs rechnerisch)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
65 +(% class="abc" %)
66 +Ermittle die Lösung der Gleichung {{formula}} 2^x = 5 {{/formula}} graphisch und rechnerisch.
67 +{{/aufgabe}}
68 +
69 +{{aufgabe id="Exponentialgleichungen Lösbarkeit (graphisch vs rechnerisch)" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="6"}}
70 +(% class="abc" %)
71 +Gegeben sind die beiden Gleichungen {{formula}} 2^x = y_0 \qquad x^2 = y_0 {{/formula}}. Untersuche ihre Lösbarkeit in Abhängigkeit von {{formula}} y_0 {{/formula}}.
72 +{{/aufgabe}}
73 +
74 +{{aufgabe id="Exponentialgleichungen (Logarithmieren)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="15"}}
75 +Bestimme die Lösungsmenge der Exponentialgleichung:
76 +(% class="abc" %)
77 +1. {{formula}} 4\cdot 0,5^x=100 {{/formula}}
78 +1. {{formula}} e^x=3 {{/formula}}
79 +1. {{formula}} 2e^x-4=8 {{/formula}}
80 +1. {{formula}} 2e^{-0.5x}=6{{/formula}}
81 +1. {{formula}} e^x=-5 {{/formula}}
82 +{{/aufgabe}}
83 +
84 +{{aufgabe id="Exponentialgleichungen (Ausklammern, SVNP)" afb="II" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="12"}}
85 +Bestimme die Lösungsmenge der Gleichung:
86 +(% class="abc" %)
87 +1. {{formula}} 2x=x^{2} {{/formula}}
88 +1. {{formula}} 2x^e=x^{2e} {{/formula}}
89 +1. {{formula}} 2e^x=e^{2x} {{/formula}}
90 +{{/aufgabe}}
91 +
92 +{{aufgabe id="Exponentialgleichungen (Substitution)" afb="III" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="12"}}
93 +Bestimme die Lösungsmenge der Gleichung:
94 +(% class="abc" %)
95 +1. {{formula}} 2x-3=x^{2} {{/formula}}
96 +1. {{formula}} 2x^e-3=x^{2e} {{/formula}}
97 +1. {{formula}} 2e^x-3=e^{2x} {{/formula}}
98 +1. {{formula}} 2e^{x-3}=e^{2x-3} {{/formula}}
99 +{{/aufgabe}}
100 +
101 +{{aufgabe id="Exponentialgleichungen" afb="I" kompetenzen="K5" quelle="Niklas Wunder" cc="BY-SA" zeit="5"}}
102 +Bestimme die Lösungsmenge der folgenden Exponentialgleichungen
103 +(% class="abc" %)
104 +1. {{formula}} 3^{x+1}=81 {{/formula}}
105 +1. {{formula}} 5^{2x}=25^{2x+2} {{/formula}}
106 +1. {{formula}} 10^{x}=500{{/formula}}
107 +1. {{formula}} 2^{x+3}=4^{x-1} {{/formula}}
108 +{{/aufgabe}}
109 +
110 +{{aufgabe id="Exponentialgleichungen graphisch" afb="II" kompetenzen="K4,K6" quelle="Niklas Wunder" cc="BY-SA" zeit="5"}}
111 +Löse mit Hilfe der nebenstehenden Abbildung folgende Exponentialgleichungen näherungsweise. Hinweis: Ordne die linke und die rechte Seite der jeweiligen Gleichung passend den Funktionsgraphen zu.
112 +(% class="abc" %)
113 +a) {{formula}} 2^x=(\frac{3}{4})^x+2 {{/formula}}
114 +b) {{formula}} 7-e^{x-3}=(\frac{3}{4})^x+2 {{/formula}}
115 +c) {{formula}} 2^x=1{,}5^{x+2}-0{,}5 {{/formula}}
116 +d) {{formula}} 7-e^{x-3}=4-\frac{1}{2}\,x {{/formula}}
117 +
118 +[[image:ExpGlei.svg]]
119 +{{/aufgabe}}
120 +
121 +{{seitenreflexion/}}
2^x und 8.svg
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38 +
ExpGlei.svg
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Logarithmus.svg
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SchaubilderExp.ggb
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1 +XWiki.niklaswunder
Größe
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x^3 und 8.svg
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1 +XWiki.martinrathgeb
Größe
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