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1 -XWiki.holgerengels
1 +XWiki.martina
Inhalt
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1 -{{seiteninhalt/}}
1 +{{box cssClass="floatinginfobox" title="**Contents**"}}
2 +{{toc start=2 depth=2 /}}
3 +{{/box}}
2 2  
5 +=== Kompetenzen ===
6 +
3 3  [[Kompetenzen.K5]] Ich kann Logarithmus nutzen, um eine Exponentialgleichung zu lösen
4 4  [[Kompetenzen.K5]] Ich kann eine geeignete Strategie wählen, um eine gegebene Exponentialgleichung zu lösen
5 5  [[Kompetenzen.K1]] Ich kann die Wahl einer Lösungsstrategie für eine Exponentialgleichung begründen
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6 6  [[Kompetenzen.K5]] Ich kann Exponentialgleichungen algebraisch lösen
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 -
10 -{{aufgabe id="Exponentialgleichungen (Logarithmieren)" afb="I" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="15"}}
11 -Bestimme die Lösungsmenge der Exponentialgleichung:
12 -(% class="abc" %)
13 -
14 -1. {{formula}} e^x=3 {{/formula}}
15 -1. {{formula}} 2e^x-4=8 {{/formula}}
16 -1. {{formula}} 2e^{-0.5x}=6{{/formula}}
17 -1. {{formula}} e^x=-5 {{/formula}}
18 -1. {{formula}} 4\cdot 5^x=100 {{/formula}}
19 -{{/aufgabe}}
20 -
21 -{{aufgabe id="Exponentialgleichungen (Satz vom Nullprodukt)" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="12"}}
22 -Bestimme die Lösungsmenge der Gleichung:
23 -(% class="abc" %)
24 -1. {{formula}} 2x-x^{2}=0 {{/formula}}
25 -1. {{formula}} 2e^x-e^{2x}=0 {{/formula}}
26 -1. {{formula}} \frac{1}{3}e^x=e^{2x} {{/formula}}
27 -1. {{formula}} 3e^{-x}=2e^{2x} {{/formula}}
28 -1. {{formula}} 2x^e=x^{2e} {{/formula}}
29 -{{/aufgabe}}
30 -
31 -{{aufgabe id="Exponentialgleichungen (Substitution)" afb="I" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="12"}}
32 -Bestimme die Lösungsmenge der Gleichung:
33 -(% class="abc" %)
34 -1. {{formula}} x^{2}-2x-3=0 {{/formula}}
35 -1. {{formula}} e^{2x}-2e^x-3=0 {{/formula}}
36 -1. {{formula}} e^x-2e^{\frac{1}{2}x}-3=0 {{/formula}}
37 -1. {{formula}} e^x-2-\frac{8}{e^x}}=0 {{/formula}}
38 -1. {{formula}} 2e^{4x}=e^{2x}+3 {{/formula}}
39 -{{/aufgabe}}
40 -
41 -{{aufgabe id="Logarithmen auswerten" afb="II" kompetenzen="K4,K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="10"}}
42 -Ordne (ohne WTR!) die Terme ihren Werten gemäß den Kästchen über dem Zahlenstrahl zu. Trage dafür die jeweiligen Buchstaben in die Kästchen ein.
43 -
44 -[[image:Logarithmus_neu.svg||width="600px"]]
45 -
46 -(% class="abc" %)
47 -1. {{formula}} \log_{10}(0.1) {{/formula}}
48 -1. {{formula}} \log_{100}(0.1) {{/formula}}
49 -1. {{formula}} \log_{0.1}(0.1) {{/formula}}
50 -1. {{formula}} \log_{10}(1000) {{/formula}}
51 -1. {{formula}} \log_{10}(50) {{/formula}}
52 -1. {{formula}} \log_{0.1}(1000) {{/formula}}
53 -1. {{formula}} \log_{10}(1) {{/formula}}
54 -1. {{formula}} \log_{100}(10) {{/formula}}
55 -1. {{formula}} \log_{10}(10) {{/formula}}
56 -{{/aufgabe}}
57 -
58 -{{aufgabe id="Exponentialgleichungen lösen (graphisch versus rechnerisch)" afb="I" kompetenzen="K4,K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="5"}}
59 -(% class="abc" %)
60 -Ermittle die Lösung der Gleichung {{formula}} e^x = 5 {{/formula}} graphisch und rechnerisch.
61 -{{/aufgabe}}
62 -
63 -{{aufgabe id="Gleichungen aufstellen I" afb="II" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe, Martina Wagner" cc="BY-SA" zeit="5"}}
64 -Nenne jeweils eine passende Gleichung:
65 -
66 -Die Gleichung kann ich nach x auflösen, indem ich …
67 -(% class="abc" %)
68 -1. … die Terme auf beiden Seiten durch 5 dividiere und damit die Lösung {{formula}} x = \frac{2}{5} {{/formula}} erhalte.
69 -1. … von beiden Termen die 5-te Wurzel ziehe und damit die Lösung {{formula}} x = \sqrt[5]{2} {{/formula}} erhalte.
70 -1. … die Terme auf beiden Seiten zur Basis 5 logarithmiere und damit die Lösung {{formula}} x = \log_5(2) {{/formula}} erhalte.
71 -{{/aufgabe}}
72 -
73 -{{aufgabe id="Darstellungen zuordnen" afb="II" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="6"}}
74 -Ordne zu:
75 -(% class="border slim" %)
76 -|Implizite Gleichungen|Explizite Gleichungen|Wertetabellen|Schaubilder
77 -|{{formula}} x^{-3} = 8 {{/formula}}|{{formula}} x = \frac{1}{\sqrt[3]{8}} {{/formula}}|(((
78 -|x|0|1|2|3
79 -|y|1|2|4|8
80 -)))|[[image:2^xund8.svg||width="200px"]]
81 -|{{formula}} 2^x = 8 {{/formula}}|{{formula}} x = -\log_{2}(8) {{/formula}} |(((
82 -|x|0|1|2|3
83 -|y|n.d.|1|{{formula}}\frac{1}{8}{{/formula}}|{{formula}}\frac{1}{27}{{/formula}}
84 -)))|[[image:2^-xund8.svg||width="200px"]]
85 -|{{formula}} 2^{-x} = 8 {{/formula}}|{{formula}} x = \log_{2}(8) {{/formula}} |(((
86 -|x|0|1|2|3
87 -|y|1|{{formula}}\frac{1}{2}{{/formula}}|{{formula}}\frac{1}{4}{{/formula}}|{{formula}}\frac{1}{8}{{/formula}}
88 -)))|[[image:x^-3und8.svg||width="200px"]]
89 -{{/aufgabe}}
90 -
91 -{{aufgabe id="Gleichungen gemeinsamer Form" afb="III" kompetenzen="K5" quelle="Martin Rathgeb" cc="BY-SA" zeit="6"}}
92 -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.
93 -(%class="abc"%)
94 -1. (((
95 -(%class="border slim"%)
96 -|(%align="center" width="160"%){{formula}}e^{2x}-4e^x+3=0{{/formula}}
97 -
98 -{{formula}}u:=\_\_\_{{/formula}}
99 -⬊|(%align="center" width="160"%){{formula}}x^{2e}-4x^e+3=0{{/formula}}
100 -
101 -{{formula}}u:=\_\_\_{{/formula}}
102 -🠗|(%align="center" width="160"%){{formula}}x^{-2}-4x^{-1}+3=0{{/formula}}
103 -
104 -{{formula}}u:=\_\_\_{{/formula}}
105 -⬋
106 -||(%align="center"%){{formula}}u^2-4u+3=0{{/formula}}
107 -(((
108 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
109 -|
110 -
111 -
112 -)))
113 -
114 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
115 -|(%align="center"%)(((⬋
116 -{{formula}}\_\_\_:=u{{/formula}}
117 -(((
118 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
119 -|
120 -
121 -
122 -)))
123 -)))|(%align="center"%)(((🠗
124 -{{formula}}\_\_\_:=u{{/formula}}
125 -(((
126 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
127 -|
128 -
129 -
130 -)))
131 -)))|(%align="center"%)(((⬊
132 -{{formula}}\_\_\_:=u{{/formula}}
133 -(((
134 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
135 -|
136 -
137 -
138 -)))
139 -)))
140 -)))
141 -1. (((
142 -(%class="border slim"%)
143 -|(%align="center" width="160"%){{formula}}x^{-2}-3x^{-1}=0{{/formula}}
144 -
145 -{{formula}}u:=\_\_\_{{/formula}}
146 -⬊|(%align="center" width="160"%){{formula}}x^{2e}-3x^e=0{{/formula}}
147 -
148 -{{formula}}u:=\_\_\_{{/formula}}
149 -🠗|(%align="center" width="160"%){{formula}}e^{2x}-3e^x=0{{/formula}}
150 -
151 -{{formula}}u:=\_\_\_{{/formula}}
152 -⬋
153 -||(%align="center"%){{formula}}u^2-3u=0{{/formula}}
154 -(((
155 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
156 -|
157 -
158 -
159 -)))
160 -
161 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
162 -|(%align="center"%)(((⬋
163 -{{formula}}\_\_\_:=u{{/formula}}
164 -(((
165 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
166 -|
167 -
168 -
169 -)))
170 -)))|(%align="center"%)(((🠗
171 -{{formula}}\_\_\_:=u{{/formula}}
172 -(((
173 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
174 -|
175 -
176 -
177 -)))
178 -)))|(%align="center"%)(((⬊
179 -{{formula}}\_\_\_:=u{{/formula}}
180 -(((
181 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
182 -|
183 -
184 -
185 -)))
186 -)))
187 -)))
188 -1. (((
189 -(%class="border slim"%)
190 -|(%align="center" width="160"%){{formula}}x^{-2}-2x^{-1}+3=0{{/formula}}
191 -
192 -{{formula}}u:=\_\_\_{{/formula}}
193 -⬊|(%align="center" width="160"%){{formula}}x^{2e}-2x^e+3=0{{/formula}}
194 -
195 -{{formula}}u:=\_\_\_{{/formula}}
196 -🠗|(%align="center" width="160"%){{formula}}e^{2x}-2e^x+3=0{{/formula}}
197 -
198 -{{formula}}u:=\_\_\_{{/formula}}
199 -⬋
200 -||(%align="center"%){{formula}}u^2-2u+3=0{{/formula}}
201 -(((
202 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
203 -|
204 -
205 -
206 -)))
207 -
208 -{{formula}}u_1=\_\_\_\quad;\quad u_2=\_\_\_{{/formula}}|
209 -|(%align="center"%)(((⬋
210 -{{formula}}\_\_\_:=u{{/formula}}
211 -(((
212 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
213 -|
214 -
215 -
216 -)))
217 -)))|(%align="center"%)(((🠗
218 -{{formula}}\_\_\_:=u{{/formula}}
219 -(((
220 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
221 -|
222 -
223 -
224 -)))
225 -)))|(%align="center"%)(((⬊
226 -{{formula}}\_\_\_:=u{{/formula}}
227 -(((
228 -(%class="border slim" style="width: 100%; margin-bottom: 0px"%)
229 -|
230 -
231 -
232 -)))
233 -)))
234 -)))
235 -{{/aufgabe}}
236 -
237 -{{aufgabe id="Gleichungstypen einstudieren" afb="III" kompetenzen="K5" quelle="Elke Hallmann, Martin Rathgeb, Dirk Tebbe, Martina Wagner" cc="BY-SA" zeit="20"}}
238 -Bestimme die Lösung der folgenden Gleichungen:
239 -
240 -(% class="border slim " %)
241 -|Typ 1 (Umkehroperationen)|Typ 2 (Ausklammern)|Typ 3 (Substitution)
242 -|{{formula}}x^2 = 2{{/formula}}|{{formula}}x^2-2x = 0{{/formula}}|{{formula}}x^4-40x^2+144 = 0{{/formula}}
243 -|{{formula}}x^4 = e{{/formula}}|{{formula}}2x^e = x^{2e}{{/formula}}|{{formula}}x^{2e}+x^e+1 = 0{{/formula}}
244 -|{{formula}}e^x = e{{/formula}}|{{formula}}2e^x = e^{2x}{{/formula}}|{{formula}}10^{6x}-2\cdot 10^{3x}+1 = 0{{/formula}}
245 -|{{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}}
246 -{{/aufgabe}}
247 -
248 -{{aufgabe id=" Exponentialgleichungen rückwärts lösen" afb="II" kompetenzen="K2,K5" quelle="Martina Wagner" lizenz="BY-SA"}}
249 -(% class="abc" %)
250 -1. ((({{{ }}}
251 -
252 -{{formula}}
253 -\begin{align*}
254 -\square e^x-2 &= 0\\
255 -\square e^x &=\square\quad \left|:\square\\
256 -e^x &= \square \\
257 -x &= 0
258 -\end{align*}
259 -{{/formula}}
260 -)))
261 -1. ((({{{ }}}
262 -
263 -{{formula}}
264 -\begin{align*}
265 -e^{2x}-\square e^x &= 0 \\
266 -e^x \cdot (\square-\square) &= 0 \left|\left| \text{ SVNP }
267 -\end{align*}
268 -{{/formula}}
269 -
270 -{{formula}}
271 -e^x \neq 0 ~und~ e^x-\square = 0{{/formula}}
272 -{{formula}} e^x=\square {{/formula}}
273 -{{formula}} x =\square {{/formula}}
274 -)))
275 -1. ((({{{ }}}
276 -
277 -{{formula}}
278 -\begin{align*}
279 -e^{2x}-\square e^x+\square &= 0 \quad \left|\left|\text{ Subst.: } e^x:=\square\\
280 -z^2-\square z + \square &= 0 \quad \left|\left|\text{ Mitternachtsformel/abc-Formel } &
281 -\end{align*}
282 -{{/formula}}
283 -
284 -{{formula}}
285 -\begin{align*}
286 -\Rightarrow z_{1,2}&=\frac{\square\pm\sqrt{\square^2-4\cdot\square\cdot\square}}{2\cdot\square}\\
287 -z_{1,2}&=\frac{\square+\square}{\square}
288 -\end{align*}
289 -{{/formula}}
290 -
291 -{{formula}}
292 -\begin{align*}
293 -&\text{Resubst.: } z:= e^x\\
294 -&e^x=\square \Rightarrow x \approx 0,693147...\\
295 -\end{align*}
296 -{{/formula}}
297 -)))
298 -{{/aufgabe}}
299 -
300 -{{aufgabe id="Gleichungen aufstellen II" afb="III" kompetenzen="K2,K5" quelle="Martin Rathgeb, Dirk Tebbe" cc="BY-SA" zeit="10"}}
301 -Nenne möglichst viele (wahre) Gleichungen der folgenden Formen, wobei {{formula}} a, b, c \in \{2; 3; 4; \ldots; 16\} {{/formula}} gelten soll:
302 -{{formula}} c = a^b\:; \qquad c = \sqrt[a]{b}\:; \qquad c = \log_a(b)\:; \qquad c = a\cdot b\:. {{/formula}}
303 -{{/aufgabe}}
304 -
305 -{{seitenreflexion/}}
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2^xund8.ggb
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