Question

Consider f(x)=4-e^x

Answer 1

$\bf \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad a^{log_a x}=x \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf f(x) = 4-e^x \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{when it crosses the x-axis, y = 0~\hfill }}{0 = 4-e^x\implies e^x=4\implies \ln(e^x)=\ln(4)\implies x=\ln(4)}~\hfill \boxed{(\stackrel{x_1}{\ln(4)}~~,~~\stackrel{y_1}{0})} \\\\[-0.35em] ~\dotfill\\\\ A)\\\\ \left. \cfrac{df}{dx}=0-e^x\right|_{x= \ln(4)}\implies -e^{ln(4)}\implies -e^{\log_e(4)}\implies -4 \\\\[-0.35em] ~\dotfill\\\\ B)\\\\ y-0=-4[x-\ln(4)]\implies y=-4x+4\ln(4)$