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Zuletzt geändert von Holger Engels am 2025/03/13 07:51
Von Version 129.1
bearbeitet von Holger Engels
am 2025/03/12 20:43
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Auf Version 77.1
bearbeitet von Martin Rathgeb
am 2025/02/26 11:39
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1 -XWiki.holgerengels
1 +XWiki.martinrathgeb
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7 7  [[Kompetenzen.K4]] [[Kompetenzen.K6]] Ich kann die Lösungen einer Exponentialgleichung als Nullstelle interpretieren
8 8  [[Kompetenzen.K4]] [[Kompetenzen.K6]] Ich kann die Lösungen einer Exponentialgleichung als Schnittstelle zweier Funktionen interpretieren
9 9  
10 -{{lehrende}}
11 11  Aufgaben:
12 12  – Logarithmus: graphisches Ermitteln vs. Operator
13 13  Lösen von Exponentialgleichungen:
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18 18  - Näherungslösungen
19 19  
20 20  Gleichungen:
21 -{{formula}}x\pm y = e \Rightarrow y = e \mp x{{/formula}}
22 -{{formula}}x*y = e \Rightarrow y = e / x{{/formula}}
23 -{{formula}}e^y = x \Rightarrow y = \ln(x){{/formula}}
24 -{{/lehrende}}
20 +x+y = e --> y = e - x
21 +x*y = e --> y = e / x
22 +e^y = x --> y = ln(x)
25 25  
26 -{{aufgabe id="Gleichungen aufstellen I" afb="II" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe, Martina Wagner" cc="BY-SA" zeit="5"}}
27 -Nenne jeweils eine passende Gleichung:
28 -
29 -Die Gleichung kann ich nach x auflösen, indem ich …
24 +{{aufgabe id="Exponentialgleichungen lösen (Fehlvorstellungen)" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="5"}}
25 +Nenne eine passende Gleichung. Die Gleichung kann ich nach x auflösen, indem ich {{formula}} \ldots {{/formula}}
30 30  (% class="abc" %)
31 -1. … die Terme auf beiden Seiten durch 5 dividiere und damit die Lösung {{formula}} x = \frac{2}{5} {{/formula}} erhalte.
32 -1. … von beiden Termen die 5-te Wurzel ziehe und damit die Lösung {{formula}} x = \sqrt[5]{2} {{/formula}} erhalte.
33 -1. … die Terme auf beiden Seiten zur Basis 5 logarithmiere und damit die Lösung {{formula}} x = \log_5(2) {{/formula}} erhalte.
27 +1. {{formula}} \ldots {{/formula}} die Terme auf beiden Seiten durch 5 dividiere und damit die Lösung {{formula}} x = \frac{2}{5} {{/formula}} erhalte.
28 +1. {{formula}} \ldots {{/formula}} von beiden Termen die 5-te Wurzel ziehe und damit die Lösung {{formula}} x = \sqrt[5]{2} {{/formula}} erhalte.
29 +1. {{formula}} \ldots {{/formula}} die Terme auf beiden Seiten zur Basis 5 logarithmiere und damit die Lösung {{formula}} x = \log_5{2} {{/formula}} erhalte.
34 34  {{/aufgabe}}
35 35  
36 -{{aufgabe id="Gleichungen aufstellen II" afb="I" kompetenzen="K2,K5" quelle="Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="10"}}
32 +{{aufgabe id="Gleichungsformen instantiieren" afb="I" kompetenzen="K2,K5" quelle="Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
37 37  Nenne möglichst viele (wahre) Gleichungen der folgenden Formen, wobei {{formula}} a, b, c \in \{2; 3; 4; \ldots; 16\} {{/formula}} gelten soll:
38 -{{formula}} c = a^b\:; \qquad c = \sqrt[a]{b}\:; \qquad c = \log_a(b)\:; \qquad c = a\cdot b\:. {{/formula}}
34 +{{formula}} c = a^b\:; \qquad c = \sqrt[a]{b}\:; \qquad c = \log_a(b)\:. {{/formula}}
39 39  {{/aufgabe}}
40 40  
41 -{{aufgabe id="Darstellungen zuordnen" afb="II" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="6"}}
42 -Ordne zu:
43 -(% class="border slim" %)
44 -|Implizite Gleichungen|Explizite Gleichungen|Wertetabellen|Schaubilder
45 -|{{formula}} x^{-3} = 8 {{/formula}}|{{formula}} x = \sqrt[3]{8} {{/formula}}|(((
46 -|x|0|1|2|3
47 -|y|1|2|4|8
48 -)))|[[image:2^xund8.svg||width="200px"]]
49 -|{{formula}} 2^x = 8 {{/formula}}|{{formula}} x = -\log_{2}(8) {{/formula}} |(((
50 -|x|0|1|2|3
51 -|y|0|1|8|27
52 -)))|[[image:2^-xund8.svg||width="200px"]]
53 -|{{formula}} 2^{-x} = 8 {{/formula}}|{{formula}} x = \log_{2}(8) {{/formula}} |(((
54 -|x|0|1|2|3
55 -|y|1|{{formula}}\frac{1}{2}{{/formula}}|{{formula}}\frac{1}{4}{{/formula}}|{{formula}}\frac{1}{8}{{/formula}}
56 -)))|[[image:x^3und8.svg||width="200px"]]
57 -|{{formula}} 2^x = 8 {{/formula}}|{{formula}} x = x = \frac{1}{\sqrt[3]{8}} {{/formula}} |(((
58 -|x|0|1|2|3
59 -|y|n.d.|1|{{formula}}\frac{1}{8}{{/formula}}|{{formula}}\frac{1}{27}{{/formula}}
60 -)))|[[image:x^-3und8.svg||width="200px"]]
37 +{{aufgabe id="Darstellungen zuordnen" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
38 +Ordne zu!
39 +(% class="abc" %)
40 +1. vier Gleichungen
41 +1. zwei Tabellen
42 +1. zwei Graphen
61 61  {{/aufgabe}}
62 62  
63 63  {{aufgabe id="Logarithmen auswerten" afb="II" kompetenzen="K4,K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="10"}}
... ... @@ -66,178 +66,28 @@
66 66  [[image:Logarithmus_neu.svg||width="600px"]]
67 67  
68 68  (% class="abc" %)
69 -1. {{formula}} \log_{10}(0.1) {{/formula}}
70 -1. {{formula}} \log_{100}(0.1) {{/formula}}
71 -1. {{formula}} \log_{0.1}(0.1) {{/formula}}
51 +1. {{formula}} \log_{10}(10) {{/formula}}
52 +1. {{formula}} \log_{100}(10) {{/formula}}
53 +1. {{formula}} \log_{11}(10) {{/formula}}
72 72  1. {{formula}} \log_{10}(1000) {{/formula}}
73 73  1. {{formula}} \log_{10}(50) {{/formula}}
74 -1. {{formula}} \log_{0.1}(1000) {{/formula}}
56 +1. {{formula}} \log_{11}(1000) {{/formula}}
75 75  1. {{formula}} \log_{10}(1) {{/formula}}
76 76  1. {{formula}} \log_{100}(10) {{/formula}}
77 77  1. {{formula}} \log_{10}(10) {{/formula}}
78 78  {{/aufgabe}}
79 79  
80 -{{aufgabe id="Exponentialgleichungen lösen (graphisch versus rechnerisch)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
62 +{{aufgabe id="Exponentialgleichungen lösen (graphisch vs rechnerisch)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
81 81  (% class="abc" %)
82 82  Ermittle die Lösung der Gleichung {{formula}} 2^x = 5 {{/formula}} graphisch und rechnerisch.
83 83  {{/aufgabe}}
84 84  
85 -{{aufgabe id="Gleichungen gemeinsamer Form" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="6"}}
86 -(%class="abc"%)
87 -1. (((
88 -(%class="border slim"%)
89 -|(%align="center"%){{formula}}x^{-2}-4x^{-1}+3=0{{/formula}}
90 -
91 -{{formula}}u:=\_\_\_{{/formula}}
92 -⬊|(%align="center"%){{formula}}x^{2e}-4x^e+3=0{{/formula}}
93 -
94 -{{formula}}u:=\_\_\_{{/formula}}
95 -🠗|(%align="center"%){{formula}}e^{2x}-4e^x+3=0{{/formula}}
96 -
97 -{{formula}}u:=\_\_\_{{/formula}}
98 -⬋
99 -||(%align="center"%){{formula}}u^2-4u+3=0{{/formula}}
100 -(((
101 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
102 -|
103 -
104 -
105 -)))
106 -
107 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
108 -|(%align="center"%)(((⬋
109 -{{formula}}\_\_\_:=u{{/formula}}
110 -(((
111 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
112 -|
113 -
114 -
115 -)))
116 -)))|(%align="center"%)(((🠗
117 -{{formula}}\_\_\_:=u{{/formula}}
118 -(((
119 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
120 -|
121 -
122 -
123 -)))
124 -)))|(%align="center"%)(((⬊
125 -{{formula}}\_\_\_:=u{{/formula}}
126 -(((
127 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
128 -|
129 -
130 -
131 -)))
132 -)))
133 -)))
134 -1. (((
135 -(%class="border slim"%)
136 -|(%align="center"%){{formula}}x^{-2}-3x^{-1}=0{{/formula}}
137 -
138 -{{formula}}u:=\_\_\_{{/formula}}
139 -⬊|(%align="center"%){{formula}}x^{2e}-3x^e=0{{/formula}}
140 -
141 -{{formula}}u:=\_\_\_{{/formula}}
142 -🠗|(%align="center"%){{formula}}e^{2x}-3e^x=0{{/formula}}
143 -
144 -{{formula}}u:=\_\_\_{{/formula}}
145 -⬋
146 -||(%align="center"%){{formula}}u^2-3u=0{{/formula}}
147 -(((
148 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
149 -|
150 -
151 -
152 -)))
153 -
154 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
155 -|(%align="center"%)(((⬋
156 -{{formula}}\_\_\_:=u{{/formula}}
157 -(((
158 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
159 -|
160 -
161 -
162 -)))
163 -)))|(%align="center"%)(((🠗
164 -{{formula}}\_\_\_:=u{{/formula}}
165 -(((
166 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
167 -|
168 -
169 -
170 -)))
171 -)))|(%align="center"%)(((⬊
172 -{{formula}}\_\_\_:=u{{/formula}}
173 -(((
174 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
175 -|
176 -
177 -
178 -)))
179 -)))
180 -)))
181 -1. (((
182 -(%class="border slim"%)
183 -|(%align="center"%){{formula}}x^{-2}-2x^{-1}+3=0{{/formula}}
184 -
185 -{{formula}}u:=\_\_\_{{/formula}}
186 -⬊|(%align="center"%){{formula}}x^{2e}-2x^e+3=0{{/formula}}
187 -
188 -{{formula}}u:=\_\_\_{{/formula}}
189 -🠗|(%align="center"%){{formula}}e^{2x}-2e^x+3=0{{/formula}}
190 -
191 -{{formula}}u:=\_\_\_{{/formula}}
192 -⬋
193 -||(%align="center"%){{formula}}u^2-2u+3=0{{/formula}}
194 -(((
195 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
196 -|
197 -
198 -
199 -)))
200 -
201 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
202 -|(%align="center"%)(((⬋
203 -{{formula}}\_\_\_:=u{{/formula}}
204 -(((
205 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
206 -|
207 -
208 -
209 -)))
210 -)))|(%align="center"%)(((🠗
211 -{{formula}}\_\_\_:=u{{/formula}}
212 -(((
213 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
214 -|
215 -
216 -
217 -)))
218 -)))|(%align="center"%)(((⬊
219 -{{formula}}\_\_\_:=u{{/formula}}
220 -(((
221 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
222 -|
223 -
224 -
225 -)))
226 -)))
227 -)))
67 +{{aufgabe id="Exponentialgleichungen Lösbarkeit (graphisch vs rechnerisch)" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="6"}}
68 +(% class="abc" %)
69 +Gegeben sind die beiden Gleichungen {{formula}} x^2 = a {{/formula}} und {{formula}} 2^x = a {{/formula}} für {{formula}} a \in \mathbb{R} {{/formula}}. Untersuche ihre Lösbarkeit in Abhängigkeit von {{formula}} a {{/formula}}.
70 +{{formula}} c = a^b\:; \qquad c = \sqrt[a]{b}\:; \qquad c = \log_a(b)\:. {{/formula}}
228 228  {{/aufgabe}}
229 229  
230 -{{aufgabe id="Gleichungstypen einstudieren" afb="II" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe, Martina Wagner" cc="BY-SA" zeit="20"}}
231 -Bestimme die Lösung der folgenden Gleichungen:
232 -
233 -(% class="border slim " %)
234 -|Typ 1 (Umkehroperationen)|Typ 2 (Ausklammern)|Typ 3 (Substitution)
235 -|{{formula}}x^2 = 2{{/formula}}|{{formula}}x^2-2x = 0{{/formula}}|{{formula}}x^4-40x^2+144 = 0{{/formula}}
236 -|{{formula}}x^4 = e{{/formula}}|{{formula}}2x^e = x^{2e}{{/formula}}|{{formula}}x^{2x}+x^e+1 = 0{{/formula}}
237 -|{{formula}}e^x = e{{/formula}}|{{formula}}2e^x = e^{2x}{{/formula}}|{{formula}}10^{6x}-2\cdot 10^{3x}+1 = 0{{/formula}}
238 -|{{formula}}3e^x = \frac{1}{2}e^{-x}{{/formula}}|{{formula}}x\cdot 3^x+4\cdot 3^x = 0{{/formula}}|{{formula}}3e^x-1 = \frac{1}{3}e^{-x}{{/formula}}
239 -{{/aufgabe}}
240 -
241 241  {{aufgabe id="Exponentialgleichungen (Logarithmieren)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="15"}}
242 242  Bestimme die Lösungsmenge der Exponentialgleichung:
243 243  (% class="abc" %)
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x^3 und 8.svg
Author
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1 +XWiki.martinrathgeb
Größe
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1 +6.7 KB
Inhalt
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