X Herausfinden (E-Funktion)

Question

0,0891*\( e^{-0,0044*x} \) = 0,019602*\( e^{0,00004*x} \)

Wie komme ich auf x = 341,02?

Answer 1

0.0891·e^(- 0.0044·x) = 0.019602·e^(0.00004·x)
e^(- 0.0044·x) / e^(0.00004·x) = 0.019602/0.0891
e^(- 0.0044·x - 0.00004·x) = 11/50
e^(- 0.00444·x) = 11/50
- 0.00444·x = LN(11/50)
x = LN(11/50) / (- 0.00444) = 341.0197596

Answer 2

Aloha :)

$$\left.0,0891\cdot e^{-0,0044x} = 0,019602\cdot e^{0,00004x} \quad\right|\;:0,019602$$$$\left.\frac{0,0891}{0,019602}\cdot e^{-0,0044x} =e^{0,00004x} \quad\right|\;\cdot e^{+0,0044x}\;\;;\;\;\frac{0,0891}{0,019602}=\frac{50}{11}$$$$\left.\frac{50}{11}=e^{0,00004x}\cdot e^{0,0044x} \quad\right|\;\text{Potenzgesetz }a^x\cdot a^y=a^{x+y}$$$$\left.\frac{50}{11}=e^{0,00004x+0,0044x} \quad\right|\;\text{Exponenten addieren}$$$$\left.\frac{50}{11}=e^{0,00444x} \quad\right|\;\ln(\cdots)$$$$\left.\ln\left(\frac{50}{11}\right)=0,00444x\quad\right|\;:0,00444$$$$\left.x=\frac{\ln\left(\frac{50}{11}\right)}{0,00444}\approx341,01976\quad\right.$$