Question

Prove that $f\left(x\right)=a^x,0<a<1$, is decreasing in R.

Answer 1

$\begin{array}{c}f\left(x\right)=a^{x}0<a<1isdecrea\sin ginR\\ Taking\log bothsides\\ \log f\left(x\right)=x\log a\\ \frac{f^{\prime }\left(x\right)}{a^x}=\log a\\ f^{\prime }\left(x\right)=a^x\log a\left[a^x>0x\in R\right]\\ f;\left(x\right)<0x\in R\\ \Rightarrow f\left(x\right)isdecre\sin gfun\left[\log a<0\right]because0<a<1\end{array}$