Solved – The exponential distribution belongs to the exponential family

Hint:

From Wikipedia:

A single-parameter exponential family is a set of probability distributions whose probability density function (or probability mass function, for the case of a discrete distribution) can be expressed in the form $f_X(xmidtheta) = h(x),exp!bigl[,eta(theta) cdot T(x) +A(theta),bigr]$ where $T(x),h(x),η(θ),$ and $A(θ)$ are known functions.

and

The probability density function (pdf) of an exponential distribution is $$ f(x;lambda) = begin{cases} lambda e^{-lambda x} & x ge 0, \ 0 & x < 0. end{cases}$$

Your mission, should you choose to accept it, is to find suitable $T(x),h(x),η(θ),$ and $A(θ)$ to demonstrate the exponential distribution pdf is of an exponential family form. It is not particularly difficult, especially if you spot $exp!bigl[,eta(theta) cdot T(x) +A(theta),bigr] = e^{A(theta)}e^{eta(theta) cdot T(x)}$. You can presume $lambda=theta$.