# Mathematrix: Aufgabenbeispiele/ Exponentialfunktion und Logarithmus

 Wie viel ist die gesuchte Variable in den folgenden Aufgaben? A)${\displaystyle \ e^{\lambda }=2\quad }$ B)${\displaystyle \ e^{\lambda }=6\quad }$ C)${\displaystyle \ e^{\lambda }=2^{2{,}5}\quad }$ D)${\displaystyle \ e^{\lambda }=2^{2{\frac {5}{13}}}\quad }$ E)${\displaystyle \ e^{\lambda }=2^{\lambda }\quad }$ F)${\displaystyle \ e^{\lambda }=a\ }$(gesucht: λ)${\displaystyle \quad }$ G)${\displaystyle \ e^{\lambda }=0{,}5\quad }$ H)${\displaystyle \ e^{\lambda }=a^{\lambda }\ }$(gesucht: λ)${\displaystyle \quad }$ I)${\displaystyle \ \ln b=4\quad }$ J)${\displaystyle \ \ln w=1\quad }$ K) ${\displaystyle \ln m=0\quad }$ L) ${\displaystyle \ln a=-1\quad }$ M) ${\displaystyle {\text{log}}x=-2\quad }$ N) ${\displaystyle {\text{log}}z=2\quad }$ O) ${\displaystyle _{7}{\text{log}}c=1\quad }$ P) ${\displaystyle _{7}{\text{log}}v=7\quad }$ Q) ${\displaystyle _{5}{\text{log}}r=3\quad }$

A)${\displaystyle \ e^{\lambda }=2\ \Rightarrow \ \ln e^{\lambda }=\ln 2\ \Rightarrow \ \lambda =\ln 2\approx 0{,}693\quad }$

B)${\displaystyle \ e^{\lambda }=6\ \Rightarrow \ \ln e^{\lambda }=\ln 6\ \Rightarrow \ \lambda =\ln 6\approx 1{,}792\quad }$

C)${\displaystyle \ e^{\lambda }=2^{2{,}5}\ \Rightarrow \ \ln e^{\lambda }=\ln 2^{2{,}5}\ \Rightarrow \ \lambda =\ln 2^{2{,}5}={2{,}5}\ln 2\approx 1{,}733\quad }$

D)${\displaystyle \ e^{\lambda }=2^{2{\frac {5}{13}}}\ \Rightarrow \ \ln e^{\lambda }=\ln 2^{2{\frac {5}{13}}}\ \Rightarrow \ \lambda =\ln 2^{\frac {31}{13}}={\frac {31}{13}}\ln 2\approx 1{,}652\quad }$

E)${\displaystyle \ e^{\lambda }=2^{\lambda }\ \Rightarrow \ \ln e^{\lambda }=\ln 2^{\lambda }\ \Rightarrow \ \lambda =\lambda \ln 2\ \Rightarrow \ \lambda (1-\ln 2)=0\ \Rightarrow \ \lambda =0\quad }$

F)${\displaystyle \ e^{\lambda }=a\ \Rightarrow \ \lambda =\ln a\quad }$ G)${\displaystyle \ e^{\lambda }=0{,}5\ \Rightarrow \ \lambda =\ln 0{,}5\approx -0{,}693\quad }$

H)${\displaystyle \ e^{\lambda }=a^{\lambda }\ \Rightarrow \ \lambda =\ln a^{\lambda }\ \Rightarrow \ \lambda =\lambda \ln a\ \Rightarrow \ \lambda (1-\ln a)=0\ \Rightarrow \ \lambda =0(a\neq e)\quad }$

I)${\displaystyle \ \ln b=4\ \Rightarrow \ b=e^{4}\approx 54{,}60\qquad }$ J)${\displaystyle \ \ln w=1\ \Rightarrow \ w=e^{1}=e\quad }$

K) ${\displaystyle \ln m=0\ \Rightarrow \ m=e^{0}=1\qquad }$ L) ${\displaystyle \ln a=-1\ \Rightarrow \ a=e^{-1}={\frac {1}{e}}\approx 0{,}368\quad }$

M) ${\displaystyle {\text{log}}x=-2\ \Rightarrow \ x=10^{-2}={\frac {1}{10^{2}}}=0{,}01\qquad }$ N) ${\displaystyle {\text{log}}z=2\ \Rightarrow \ z=10^{2}=100\quad }$

O) ${\displaystyle _{7}{\text{log}}c=1\Rightarrow \ c=7^{1}=7\qquad }$ P) ${\displaystyle _{7}{\text{log}}v=7\Rightarrow \ v=7^{7}=823543\quad }$

Q) ${\displaystyle _{5}{\text{log}}r=3\Rightarrow \ r=5^{3}=125\quad }$