• Exponentialgleichungen lösen
  • Simon Brückner
  • 30.06.2020
  • Mathematik
  • 11
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Exponentialgleichungen lösen

1
Logarithmieren

Erinnerung

x=loga(b)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} x=log_a(b) \gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \Leftrightarrow ax=b\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} a^x=b
x=ln(b)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} x=ln(b) \gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \Leftrightarrow ex=b\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^x=b

Lösen Sie die Gleichungen. Überprüfen Sie Ihre Ergeb-
nisse mithilfe des nebenstehenden Videos oder der
Lösungskärtchen.
  • 2x32=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2^x-32=0
  • ex4=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^x-4=0

https://vimeo.com/343912808

c) 2e4x4=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2\cdot e^{4x}-4=0

e4x=2\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{4x}=2

4x=ln(2)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 4x=ln(2)

e4x2=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{4x}-2=0

x=0,25ln(2)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} x=0{,}25\cdot ln(2)

Übung

Hinweis

Nicht jede Exponential- gleichung besitzt eine reelle Lösung!

1
Lösen Sie möglichst weit ohne Taschenrechner.
  • 2x+8=16\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2^x+8=16
  • 42x+5=21\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 4^{2x}+5=21
  • 2+ex=3\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2+e^x=3
  • 2x+8=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2^x+8=0
  • e3x=e\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{3x}=e
  • ex+3=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^x+3=0
a) x=3 b) x=1 c) x=0 d) keine Lösung e) x=1/3 f) keine Lösung

Ausklammern

vimeo.com/285266276

e2x4ex=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{2x}-4e^x=0

Substitution

vimeo.com/288848904

e2x2ex8=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{2x}-2e^x-8=0

Übung

2
Lösen Sie durch ausklammern.
  • e2x8ex=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{2x}-8e^x=0
  • e2x+8ex=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{2x}+8e^x=0
  • 4x32x=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 4^x-3\cdot2^x=0
Tipp

4=2²

3
Lösen Sie durch Substitution.
  • 2e2x6ex+4=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2e^{2x}-6e^x+4=0
  • e2x3=2ex\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{2x}-3=2e^x
  • 4x52x=4\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 4^x-5\cdot2^x=-4
4
Lösen Sie mit einem geeigneten Verfahren.
  • e2x+4ex3=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} -e^{2x}+4e^x-3=0
  • 4e3x8ex=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 4e^{3x}-8e^x=0
  • 2e2x+6ex20=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2e^{2x}+6e^x-20=0
  • e2x3ex=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} e^{2x}-3e^x=0
  • 2e2x10=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2e^{2x}-10=0
  • 24x82x=6\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2\cdot 4^x-8\cdot 2^x=-6
  • 29x3x=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 2\cdot 9^x-3^x=0
  • 8x0,252x=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} 8^x-0{,}25\cdot2^x=0
2a) {2,08} b) { } c) {1,58} 3a) {0; 0,69} b) {1,1} c) {0; 2} 4a) {0; 1,1} b) {0,35} c) {0,69} 4d) {1,1} e) {0,8} f) {0; 1,58} 4g) {-0,63} h) {-1}

Lösung