Wie viel ist die gesuchte Variable in den folgenden Aufgaben? A) e λ = 2 {\displaystyle \ e^{\lambda }=2\quad } B) e λ = 6 {\displaystyle \ e^{\lambda }=6\quad } C) e λ = 2 2 , 5 {\displaystyle \ e^{\lambda }=2^{2{,}5}\quad } D) e λ = 2 2 5 13 {\displaystyle \ e^{\lambda }=2^{2{\frac {5}{13}}}\quad } E) e λ = 2 λ {\displaystyle \ e^{\lambda }=2^{\lambda }\quad } F) e λ = a {\displaystyle \ e^{\lambda }=a\ } (gesucht: λ) {\displaystyle \quad } G) e λ = 0 , 5 {\displaystyle \ e^{\lambda }=0{,}5\quad } H) e λ = a λ {\displaystyle \ e^{\lambda }=a^{\lambda }\ } (gesucht: λ) {\displaystyle \quad } I) ln b = 4 {\displaystyle \ \ln b=4\quad } J) ln w = 1 {\displaystyle \ \ln w=1\quad } K) ln m = 0 {\displaystyle \ln m=0\quad } L) ln a = − 1 {\displaystyle \ln a=-1\quad } M) log x = − 2 {\displaystyle {\text{log}}x=-2\quad } N) log z = 2 {\displaystyle {\text{log}}z=2\quad } O) 7 log c = 1 {\displaystyle _{7}{\text{log}}c=1\quad } P) 7 log v = 7 {\displaystyle _{7}{\text{log}}v=7\quad } Q) 5 log r = 3 {\displaystyle _{5}{\text{log}}r=3\quad }
A) e λ = 2 ⇒ ln e λ = ln 2 ⇒ λ = ln 2 ≈ 0,693 {\displaystyle \ e^{\lambda }=2\ \Rightarrow \ \ln e^{\lambda }=\ln 2\ \Rightarrow \ \lambda =\ln 2\approx 0{,}693\quad } B) e λ = 6 ⇒ ln e λ = ln 6 ⇒ λ = ln 6 ≈ 1,792 {\displaystyle \ e^{\lambda }=6\ \Rightarrow \ \ln e^{\lambda }=\ln 6\ \Rightarrow \ \lambda =\ln 6\approx 1{,}792\quad } C) e λ = 2 2 , 5 ⇒ ln e λ = ln 2 2 , 5 ⇒ λ = ln 2 2 , 5 = 2 , 5 ln 2 ≈ 1,733 {\displaystyle \ e^{\lambda }=2^{2{,}5}\ \Rightarrow \ \ln e^{\lambda }=\ln 2^{2{,}5}\ \Rightarrow \ \lambda =\ln 2^{2{,}5}={2{,}5}\ln 2\approx 1{,}733\quad } D) e λ = 2 2 5 13 ⇒ ln e λ = ln 2 2 5 13 ⇒ λ = ln 2 31 13 = 31 13 ln 2 ≈ 1,652 {\displaystyle \ e^{\lambda }=2^{2{\frac {5}{13}}}\ \Rightarrow \ \ln e^{\lambda }=\ln 2^{2{\frac {5}{13}}}\ \Rightarrow \ \lambda =\ln 2^{\frac {31}{13}}={\frac {31}{13}}\ln 2\approx 1{,}652\quad } E) e λ = 2 λ ⇒ ln e λ = ln 2 λ ⇒ λ = λ ln 2 ⇒ λ ( 1 − ln 2 ) = 0 ⇒ λ = 0 {\displaystyle \ e^{\lambda }=2^{\lambda }\ \Rightarrow \ \ln e^{\lambda }=\ln 2^{\lambda }\ \Rightarrow \ \lambda =\lambda \ln 2\ \Rightarrow \ \lambda (1-\ln 2)=0\ \Rightarrow \ \lambda =0\quad } F) e λ = a ⇒ λ = ln a {\displaystyle \ e^{\lambda }=a\ \Rightarrow \ \lambda =\ln a\quad } G) e λ = 0 , 5 ⇒ λ = ln 0 , 5 ≈ − 0,693 {\displaystyle \ e^{\lambda }=0{,}5\ \Rightarrow \ \lambda =\ln 0{,}5\approx -0{,}693\quad } H) e λ = a λ ⇒ λ = ln a λ ⇒ λ = λ ln a ⇒ λ ( 1 − ln a ) = 0 ⇒ λ = 0 ( a ≠ e ) {\displaystyle \ e^{\lambda }=a^{\lambda }\ \Rightarrow \ \lambda =\ln a^{\lambda }\ \Rightarrow \ \lambda =\lambda \ln a\ \Rightarrow \ \lambda (1-\ln a)=0\ \Rightarrow \ \lambda =0(a\neq e)\quad } I) ln b = 4 ⇒ b = e 4 ≈ 54 , 60 {\displaystyle \ \ln b=4\ \Rightarrow \ b=e^{4}\approx 54{,}60\qquad } J) ln w = 1 ⇒ w = e 1 = e {\displaystyle \ \ln w=1\ \Rightarrow \ w=e^{1}=e\quad } K) ln m = 0 ⇒ m = e 0 = 1 {\displaystyle \ln m=0\ \Rightarrow \ m=e^{0}=1\qquad } L) ln a = − 1 ⇒ a = e − 1 = 1 e ≈ 0,368 {\displaystyle \ln a=-1\ \Rightarrow \ a=e^{-1}={\frac {1}{e}}\approx 0{,}368\quad } M) log x = − 2 ⇒ x = 10 − 2 = 1 10 2 = 0 , 01 {\displaystyle {\text{log}}x=-2\ \Rightarrow \ x=10^{-2}={\frac {1}{10^{2}}}=0{,}01\qquad } N) log z = 2 ⇒ z = 10 2 = 100 {\displaystyle {\text{log}}z=2\ \Rightarrow \ z=10^{2}=100\quad } O) 7 log c = 1 ⇒ c = 7 1 = 7 {\displaystyle _{7}{\text{log}}c=1\Rightarrow \ c=7^{1}=7\qquad } P) 7 log v = 7 ⇒ v = 7 7 = 823543 {\displaystyle _{7}{\text{log}}v=7\Rightarrow \ v=7^{7}=823543\quad } Q) 5 log r = 3 ⇒ r = 5 3 = 125 {\displaystyle _{5}{\text{log}}r=3\Rightarrow \ r=5^{3}=125\quad }
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