Wiki-Quellcode von Lösung Exponentialgleichungen rückwärts lösen
Zuletzt geändert von akukin am 2025/08/11 15:30
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| author | version | line-number | content |
|---|---|---|---|
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1.1 | 1 | |
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5.1 | 2 | 1. ((( |
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1.1 | 3 | |
| 4 | {{formula}} | ||
| 5 | \begin{align*} | ||
| 6 | 2 e^x-2 &= 0\\ | ||
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5.1 | 7 | 2 e^x &=2\quad \mid |:2\\ |
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1.1 | 8 | e^x &=1 \\ |
| 9 | x &= 0 | ||
| 10 | \end{align*} | ||
| 11 | {{/formula}} | ||
| 12 | ))) | ||
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5.1 | 13 | 1. ((( |
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1.1 | 14 | |
| 15 | {{formula}} | ||
| 16 | \begin{align*} | ||
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2.1 | 17 | e^{2x}-e \cdot e^x &= 0 \\ |
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5.1 | 18 | e^x\cdot(e^x-e) &= 0 \mid\mid \text{ SVNP } |
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1.1 | 19 | \end{align*} |
| 20 | {{/formula}} | ||
| 21 | |||
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2.1 | 22 | {{formula}}\ e^x \neq 0 ~und~ e^x-e = 0{{/formula}} |
| 23 | {{formula}}\ e^x = e{{/formula}} | ||
| 24 | {{formula}} x =1 {{/formula}} | ||
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1.1 | 25 | ))) |
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5.1 | 26 | 1. ((( |
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1.1 | 27 | |
| 28 | {{formula}} | ||
| 29 | \begin{align*} | ||
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5.1 | 30 | e^{2x}-4 e^x+4 &= 0 \quad \mid\mid\text{ Subst.: } e^x:=z\\ |
| 31 | z^2-4 z + 4 &= 0 \quad \mid\mid\text{ Mitternachtsformel/abc-Formel } & | ||
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1.1 | 32 | \end{align*} |
| 33 | {{/formula}} | ||
| 34 | |||
| 35 | {{formula}} | ||
| 36 | \begin{align*} | ||
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3.1 | 37 | \Rightarrow z_{1,2}&=\frac{4\pm\sqrt{4^2-4\cdot 1\cdot 4}}{2\cdot 1}\\ |
| 38 | z_{1,2}&=\frac{4+0}{2}=2 | ||
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1.1 | 39 | \end{align*} |
| 40 | {{/formula}} | ||
| 41 | |||
| 42 | {{formula}} | ||
| 43 | \begin{align*} | ||
| 44 | &\text{Resubst.: } z:= e^x\\ | ||
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5.1 | 45 | &e^x=2 \mid\mid \text{ ln } |
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4.1 | 46 | \Rightarrow x \approx 0,693147...\\ |
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1.1 | 47 | \end{align*} |
| 48 | {{/formula}} | ||
| 49 | ))) | ||
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3.1 | 50 |