Änderungen von Dokument Lösung Exponentialgleichungen (Substitution)
Zuletzt geändert von Kim Fujan am 2025/05/21 10:47
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... ... @@ -1,60 +1,13 @@ 1 1 (% class="abc" %) 2 -1. {{formula}} x^{2}-2x-3=0 {{/formula}} 3 - 2 +1. {{formula}} x^{2}+2x-3=0 {{/formula}} 4 4 Lösung mit abc-Formel: 5 5 {{formula}}x_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}} 6 6 {{formula}} x_{1}=3 \quad ; \quad x_{2}=-1 {{/formula}} 6 + 7 +1. {{formula}} e^{2x}+2e^x-3=0 {{/formula}} 7 7 8 -1. {{formula}} e^ {2x}-2e^x-3=0 {{/formula}}9 +1. {{formula}} e^x+2e^{\frac{1}{2}x}-3=0 {{/formula}} 9 9 10 -Substitution: {{formula}} e^x=u {{/formula}} 11 -{{formula}} u^{2}+2u-3=0 {{/formula}} 11 +1. {{formula}} e^x-2-\frac{15}{e^x}}=0 {{/formula}} 12 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}} 16 - 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 - 21 -1. {{formula}} e^x-2e^{\frac{1}{2}x}-3=0 {{/formula}} 22 - 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! 33 - 34 -1. {{formula}} e^x-2-\frac{8}{e^x}}=0 {{/formula}} 35 -{{formula}} e^{-x} \cdot (e^{2x}-2e^x-8)=0 {{/formula}} 36 - 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: 45 -{{formula}} e^{x}=4 \quad \Longleftrightarrow \quad x=2 \cdot ln(4) {{/formula}} 46 -{{formula}} e^{x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung! 47 - 48 48 1. {{formula}} 2e^{4x}=e^{2x}+3 {{/formula}} 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: 59 -{{formula}} e^{\frac{1}{2}x}=3 \quad \Longleftrightarrow \quad x=2 \cdot ln(3) {{/formula}} 60 -{{formula}} e^{\frac{1}{2}x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung!