Lösung Exponentialgleichungen rückwärts lösen

Version 4.1 von Martina Wagner am 2025/05/20 12:31

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\[\begin{align*} 2 e^x-2 &= 0\\ 2 e^x &=2\quad \left|:2\\ e^x &=1 \\ x &= 0 \end{align*}\]
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\[\begin{align*} e^{2x}-e \cdot e^x &= 0 \\ e^x\cdot(e^x-e) &= 0 \left|\left| \text{ SVNP } \end{align*}\]
\(\ e^x \neq 0 ~und~ e^x-e = 0\)
\(\ e^x = e\)
\( x =1 \)
```
 
```
\[\begin{align*} e^{2x}-4 e^x+4 &= 0 \quad \left|\left|\text{ Subst.: } e^x:=z\\ z^2-4 z + 4 &= 0 \quad \left|\left|\text{ Mitternachtsformel/abc-Formel } & \end{align*}\]
\[\begin{align*} \Rightarrow z_{1,2}&=\frac{4\pm\sqrt{4^2-4\cdot 1\cdot 4}}{2\cdot 1}\\ z_{1,2}&=\frac{4+0}{2}=2 \end{align*}\]
\[\begin{align*} &\text{Resubst.: } z:= e^x\\ &e^x=2 \left|\left|\text{ ln } \Rightarrow x \approx 0,693147...\\ \end{align*}\]