Wiki-Quellcode von Lösung Exponentialgleichungen (Substitution)
Zuletzt geändert von akukin am 2025/08/11 15:27
Verstecke letzte Bearbeiter
| author | version | line-number | content |
|---|---|---|---|
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2.1 | 1 | (% class="abc" %) |
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6.1 | 2 | 1. {{formula}} x^{2}-2x-3=0 {{/formula}} |
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4.1 | 3 | |
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2.1 | 4 | Lösung mit abc-Formel: |
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3.1 | 5 | {{formula}}x_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}} |
| 6 | {{formula}} x_{1}=3 \quad ; \quad x_{2}=-1 {{/formula}} | ||
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4.1 | 7 | |
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6.1 | 8 | 1. {{formula}} e^{2x}-2e^x-3=0 {{/formula}} |
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2.1 | 9 | |
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4.1 | 10 | Substitution: {{formula}} e^x=u {{/formula}} |
| 11 | {{formula}} u^{2}+2u-3=0 {{/formula}} | ||
| 12 | |||
| 13 | Lösung mit abc-Formel: | ||
| 14 | {{formula}}u_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}} | ||
| 15 | {{formula}} u_{1}=3 \quad ; \quad u_{2}=-1 {{/formula}} | ||
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5.1 | 16 | |
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4.1 | 17 | Resubstitution: |
| 18 | {{formula}} e^x=3 \quad \Longleftrightarrow \quad x=ln(3) {{/formula}} | ||
| 19 | {{formula}} e^x=-1 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung! | ||
| 20 | |||
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6.1 | 21 | 1. {{formula}} e^x-2e^{\frac{1}{2}x}-3=0 {{/formula}} |
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2.1 | 22 | |
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5.1 | 23 | Substitution: {{formula}} e^{\frac{1}{2}x}=u {{/formula}} |
| 24 | {{formula}} u^{2}+2u-3=0 {{/formula}} | ||
| 25 | |||
| 26 | Lösung mit abc-Formel: | ||
| 27 | {{formula}}u_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}} | ||
| 28 | {{formula}} u_{1}=3 \quad ; \quad u_{2}=-1 {{/formula}} | ||
| 29 | |||
| 30 | Resubstitution: | ||
| 31 | {{formula}} e^{\frac{1}{2}x}=3 \quad \Longleftrightarrow \quad x=2 \cdot ln(3) {{/formula}} | ||
| 32 | {{formula}} e^{\frac{1}{2}x}=-1 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung! | ||
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6.1 | 33 | |
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12.1 | 34 | 1. {{formula}} e^x-2-\frac{8}{e^x}=0 {{/formula}} |
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8.1 | 35 | {{formula}} e^{-x} \cdot (e^{2x}-2e^x-8)=0 {{/formula}} |
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2.1 | 36 | |
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8.1 | 37 | Substitution: {{formula}} e^{x}=u {{/formula}} |
| 38 | {{formula}} u^{2}-2u-8=0 {{/formula}} | ||
| 39 | |||
| 40 | Lösung mit abc-Formel: | ||
| 41 | {{formula}}u_{1,2}=\frac{2\pm \sqrt{4+32}}{2}=\frac{2\pm 6}{2}{{/formula}} | ||
| 42 | {{formula}} u_{1}=4 \quad ; \quad u_{2}=-2 {{/formula}} | ||
| 43 | |||
| 44 | Resubstitution: | ||
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8.2 | 45 | {{formula}} e^{x}=4 \quad \Longleftrightarrow \quad x= ln(4) {{/formula}} |
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8.1 | 46 | {{formula}} e^{x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung! |
| 47 | |||
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1.1 | 48 | 1. {{formula}} 2e^{4x}=e^{2x}+3 {{/formula}} |
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6.1 | 49 | {{formula}} 2e^{4x}-e^{2x}-3=0 {{/formula}} |
| 50 | |||
| 51 | Substitution: {{formula}} e^{2x}=u {{/formula}} | ||
| 52 | {{formula}} 2u^{2}-u-3=0 {{/formula}} | ||
| 53 | |||
| 54 | Lösung mit abc-Formel: | ||
| 55 | {{formula}}u_{1,2}=\frac{1\pm \sqrt{1+24}}{2}=\frac{1\pm 5}{2}{{/formula}} | ||
| 56 | {{formula}} u_{1}=3 \quad ; \quad u_{2}=-2 {{/formula}} | ||
| 57 | |||
| 58 | Resubstitution: | ||
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10.1 | 59 | {{formula}} e^{2x}=3 \quad \Longleftrightarrow \quad x=\frac{1}{2} \cdot ln(3) {{/formula}} |
| 60 | {{formula}} e^{2x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung! |