how do I find the inverse

Let f(x)  = y   and we have

y = 2 + e^x+4      the idea is to get x by itself and then "switch" x and y

Subtract 2 from both sides

y - 2   =  e^{x + 4}     take the natural log of both sides

ln [y - 2] = ln e ^{x + 4}       and by a log property we can write

ln [y -2] = (x + 4) ln e          ln e  = 1    so we can now ignore this

ln [y - 2]  = x + 4           subtract 4 from both sides

ln [y - 2]  -  4   = x         "switch" x and y

ln [x - 2] - 4  = y         and, for y, write  f^-1(x)     where f^-1(x)  indicates the inverse function

ln [ x - 2 ] - 4  = f^--1(x)

Here's the graph of the original function and its inverse : https://www.desmos.com/calculator/wr4hgezjxu