Chapter 4, Exponential and Logarithmic Functions - Section 4.5 - Exponential and Logarithmic Functions - 4.5 Exercises - Page 404: 38

(a) $x=-\ln 4$ (b) $x\approx -1.386294$

(a) We need to solve: $\frac{10}{1+e^{-x}}=2$ $10=2(1+e^{-x})$ $10=2+2e^{-x}$ $10-2=8=2e^{-x}$ $4=e^{-x}$ We take the natural log of both sides and use log rules to simplify: $\ln 4=\ln e^{-x}$ $\ln 4=-x$ $x=-\ln 4$ (b) We solve using a calculator: $x\approx -1.386294$