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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:
... ... @@ -18,230 +18,58 @@
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:
24 +{{aufgabe id="Exponentialgleichungen lösen (Grund- vs Fehlvorstellungen)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" 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. Operation.
30 +)))
31 +1. Umkehraufgaben: Gib für die in a) falsche Methode(n) eine passende Gleichung an.
32 +{{/aufgabe}}
28 28  
29 -Die Gleichung kann ich nach x auflösen, indem ich …
34 +{{aufgabe id="Exponentialgleichungen lösen (graphisch vs rechnerisch)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
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.
36 +Bestimme die Lösung der Gleichung {{formula}} 2^x = 5 {{/formula}} graphisch und rechnerisch.
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"}}
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}}
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
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"]]
47 +{{aufgabe id="Gleichungsformen besetzen" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
48 +Bilde für {{formula}} a, b, c \in \{2; 3; 4; \ldots; 16\} {{/formula}} möglichst viele Gleichungen der folgenden Typen:
49 +{{formula}} c = a^b\:; \qquad c = \sqrt[a]{b}\:; \qquad c = \log_a(b)\:. {{/formula}}
61 61  {{/aufgabe}}
62 62  
63 -{{aufgabe id="Logarithmen auswerten" afb="II" kompetenzen="K4,K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="10"}}
64 -Ordne (ohne WTR!) die Terme ihren Werten gemäß den Kästchen über dem Zahlenstrahl zu. Trage dafür die jeweiligen Buchstaben in die Kästchen ein.
52 +{{aufgabe id="Logarithmen auswerten" afb="II" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="10"}}
53 +Ordne (ohne WTR) die Terme ihren Werten gemäß den Kästchen über dem Zahlenstrahl zu. Trage dafür die jeweiligen Buchstaben in die Kästchen ein.
65 65  
66 -[[image:Logarithmus_neu.svg||width="600px"]]
55 +[[image:Logarithmus.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}}
58 +1. {{formula}} \log_{10}(10) {{/formula}}
59 +1. {{formula}} \log_{100}(10) {{/formula}}
60 +1. {{formula}} \log_{11}(10) {{/formula}}
72 72  1. {{formula}} \log_{10}(1000) {{/formula}}
73 -1. {{formula}} \log_{10}(50) {{/formula}}
74 -1. {{formula}} \log_{0.1}(1000) {{/formula}}
62 +1. {{formula}} \log_{10}(5) {{/formula}}
63 +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"}}
69 +{{aufgabe id="Exponentialgleichungen (Logarithmieren)" afb="I" kompetenzen="K5" quelle="Niklas Wunder" cc="BY-SA" zeit="10"}}
70 +Bestimme die Lösungsmenge der folgenden Exponentialgleichungen:
81 81  (% class="abc" %)
82 -Ermittle die Lösung der Gleichung {{formula}} 2^x = 5 {{/formula}} graphisch und rechnerisch.
83 -{{/aufgabe}}
84 -
85 -{{aufgabe id="Gleichungen gemeinsamer Form" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="6"}}
86 -Die Gleichungen sehen auf den ersten Blick unterschiedlich aus, weisen aber ähnliche Strukturen auf und können alle mithilfe der Substitution gelöst werden. Selbstverständlich gibt es für manche Teilaufgaben auch andere Lösungswege ohne Substitution.
87 -(%class="abc"%)
88 -1. (((
89 -(%class="border slim"%)
90 -|(%align="center" width="160"%){{formula}}x^{-2}-4x^{-1}+3=0{{/formula}}
91 -
92 -{{formula}}u:=\_\_\_{{/formula}}
93 -⬊|(%align="center" width="160"%){{formula}}x^{2e}-4x^e+3=0{{/formula}}
94 -
95 -{{formula}}u:=\_\_\_{{/formula}}
96 -🠗|(%align="center" width="160"%){{formula}}e^{2x}-4e^x+3=0{{/formula}}
97 -
98 -{{formula}}u:=\_\_\_{{/formula}}
99 -⬋
100 -||(%align="center"%){{formula}}u^2-4u+3=0{{/formula}}
101 -(((
102 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
103 -|
104 -
105 -
106 -)))
107 -
108 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
109 -|(%align="center"%)(((⬋
110 -{{formula}}\_\_\_:=u{{/formula}}
111 -(((
112 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
113 -|
114 -
115 -
116 -)))
117 -)))|(%align="center"%)(((🠗
118 -{{formula}}\_\_\_:=u{{/formula}}
119 -(((
120 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
121 -|
122 -
123 -
124 -)))
125 -)))|(%align="center"%)(((⬊
126 -{{formula}}\_\_\_:=u{{/formula}}
127 -(((
128 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
129 -|
130 -
131 -
132 -)))
133 -)))
134 -)))
135 -1. (((
136 -(%class="border slim"%)
137 -|(%align="center" width="160"%){{formula}}x^{-2}-3x^{-1}=0{{/formula}}
138 -
139 -{{formula}}u:=\_\_\_{{/formula}}
140 -⬊|(%align="center" width="160"%){{formula}}x^{2e}-3x^e=0{{/formula}}
141 -
142 -{{formula}}u:=\_\_\_{{/formula}}
143 -🠗|(%align="center" width="160"%){{formula}}e^{2x}-3e^x=0{{/formula}}
144 -
145 -{{formula}}u:=\_\_\_{{/formula}}
146 -⬋
147 -||(%align="center"%){{formula}}u^2-3u=0{{/formula}}
148 -(((
149 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
150 -|
151 -
152 -
153 -)))
154 -
155 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
156 -|(%align="center"%)(((⬋
157 -{{formula}}\_\_\_:=u{{/formula}}
158 -(((
159 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
160 -|
161 -
162 -
163 -)))
164 -)))|(%align="center"%)(((🠗
165 -{{formula}}\_\_\_:=u{{/formula}}
166 -(((
167 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
168 -|
169 -
170 -
171 -)))
172 -)))|(%align="center"%)(((⬊
173 -{{formula}}\_\_\_:=u{{/formula}}
174 -(((
175 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
176 -|
177 -
178 -
179 -)))
180 -)))
181 -)))
182 -1. (((
183 -(%class="border slim"%)
184 -|(%align="center" width="160"%){{formula}}x^{-2}-2x^{-1}+3=0{{/formula}}
185 -
186 -{{formula}}u:=\_\_\_{{/formula}}
187 -⬊|(%align="center" width="160"%){{formula}}x^{2e}-2x^e+3=0{{/formula}}
188 -
189 -{{formula}}u:=\_\_\_{{/formula}}
190 -🠗|(%align="center" width="160"%){{formula}}e^{2x}-2e^x+3=0{{/formula}}
191 -
192 -{{formula}}u:=\_\_\_{{/formula}}
193 -⬋
194 -||(%align="center"%){{formula}}u^2-2u+3=0{{/formula}}
195 -(((
196 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
197 -|
198 -
199 -
200 -)))
201 -
202 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
203 -|(%align="center"%)(((⬋
204 -{{formula}}\_\_\_:=u{{/formula}}
205 -(((
206 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
207 -|
208 -
209 -
210 -)))
211 -)))|(%align="center"%)(((🠗
212 -{{formula}}\_\_\_:=u{{/formula}}
213 -(((
214 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
215 -|
216 -
217 -
218 -)))
219 -)))|(%align="center"%)(((⬊
220 -{{formula}}\_\_\_:=u{{/formula}}
221 -(((
222 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
223 -|
224 -
225 -
226 -)))
227 -)))
228 -)))
229 -{{/aufgabe}}
230 -
231 -{{aufgabe id="Gleichungstypen einstudieren" afb="II" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe, Martina Wagner" cc="BY-SA" zeit="20"}}
232 -Bestimme die Lösung der folgenden Gleichungen:
233 -
234 -(% class="border slim " %)
235 -|Typ 1 (Umkehroperationen)|Typ 2 (Ausklammern)|Typ 3 (Substitution)
236 -|{{formula}}x^2 = 2{{/formula}}|{{formula}}x^2-2x = 0{{/formula}}|{{formula}}x^4-40x^2+144 = 0{{/formula}}
237 -|{{formula}}x^4 = e{{/formula}}|{{formula}}2x^e = x^{2e}{{/formula}}|{{formula}}x^{2x}+x^e+1 = 0{{/formula}}
238 -|{{formula}}e^x = e{{/formula}}|{{formula}}2e^x = e^{2x}{{/formula}}|{{formula}}10^{6x}-2\cdot 10^{3x}+1 = 0{{/formula}}
239 -|{{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}}
240 -{{/aufgabe}}
241 -
242 -{{aufgabe id="Exponentialgleichungen (Logarithmieren)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="15"}}
243 -Bestimme die Lösungsmenge der Exponentialgleichung:
244 -(% class="abc" %)
245 245  1. {{formula}} 4\cdot 0,5^x=100 {{/formula}}
246 246  1. {{formula}} e^x=3 {{/formula}}
247 247  1. {{formula}} 2e^x-4=8 {{/formula}}
... ... @@ -249,21 +249,16 @@
249 249  1. {{formula}} e^x=-5 {{/formula}}
250 250  {{/aufgabe}}
251 251  
252 -{{aufgabe id="Exponentialgleichungen (Nullproduktsatz)" afb="II" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="12"}}
253 -Bestimme die Lösungsmenge der Gleichung:
79 +{{aufgabe id="Exponentialgleichungen (Ausklammern, SVNP)" afb="II" kompetenzen="K5" quelle="Niklas Wunder" cc="BY-SA" zeit="5"}}
80 +Bestimme die Lösungsmenge der folgenden Exponentialgleichungen:
254 254  (% class="abc" %)
255 -1. {{formula}} 2x=x^{2} {{/formula}}
256 -1. {{formula}} 2x^e=x^{2e} {{/formula}}
257 257  1. {{formula}} 2e^x=e^{2x} {{/formula}}
258 258  {{/aufgabe}}
259 259  
260 -{{aufgabe id="Exponentialgleichungen (Substitution)" afb="III" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="12"}}
261 -Bestimme die Lösungsmenge der Gleichung:
85 +{{aufgabe id="Exponentialgleichungen (Substitution)" afb="III" kompetenzen="K5" quelle="Niklas Wunder" cc="BY-SA" zeit="5"}}
86 +Bestimme die Lösungsmenge der folgenden Exponentialgleichungen:
262 262  (% class="abc" %)
263 -1. {{formula}} 2x-3=x^{2} {{/formula}}
264 -1. {{formula}} 2x^e-3=x^{2e} {{/formula}}
265 265  1. {{formula}} 2e^x-3=e^{2x} {{/formula}}
266 -1. {{formula}} 2e^{x-3}=e^{2x-3} {{/formula}}
267 267  {{/aufgabe}}
268 268  
269 269  {{aufgabe id="Exponentialgleichungen" afb="I" kompetenzen="K5" quelle="Niklas Wunder" cc="BY-SA" zeit="5"}}
... ... @@ -278,12 +278,12 @@
278 278  {{aufgabe id="Exponentialgleichungen graphisch" afb="II" kompetenzen="K4,K6" quelle="Niklas Wunder" cc="BY-SA" zeit="5"}}
279 279  Löse mit Hilfe der nebenstehenden Abbildung folgende Exponentialgleichungen näherungsweise. Hinweis: Ordne die linke und die rechte Seite der jeweiligen Gleichung passend den Funktionsgraphen zu.
280 280  (% class="abc" %)
281 -1. {{formula}} 2^x=(\frac{3}{4})^x+2 {{/formula}}
282 -1. {{formula}} 7-e^{x-3}=(\frac{3}{4})^x+2 {{/formula}}
283 -1. {{formula}} 2^x=1{,}5^{x+2}-0{,}5 {{/formula}}
284 -1. {{formula}} 7-e^{x-3}=4-\frac{1}{2}\,x {{/formula}}
103 +a) {{formula}} 2^x=(\frac{3}{4})^x+2 {{/formula}}
104 +b) {{formula}} 7-e^{x-3}=(\frac{3}{4})^x+2 {{/formula}}
105 +c) {{formula}} 2^x=1{,}5^{x+2}-0{,}5 {{/formula}}
106 +d) {{formula}} 7-e^{x-3}=4-\frac{1}{2}\,x {{/formula}}
285 285  
286 -[[image:ExpGlei.svg||width="600px"]]
108 +[[image:ExpGlei.svg]]
287 287  {{/aufgabe}}
288 288  
289 289  {{seitenreflexion/}}
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