How do you solve ##ln e^x -ln e^3 = ln e^9##?

If you use the properties of logarithms, you will need to refer to these:

  1. ##lna^b = blna##
  2. ##lne = 1##

OR

##lne^z = z##

where ##a## and ##b## are constants, ##e = 2.718281828cdots##, and ##z## is either a constant or a variable. Therefore, we can do this:

##lne^x – lne^3 = lne^9##

##lne^x = lne^9 + lne^3##

##xcancel(lne) = 9cancel(lne) + 3cancel(lne)##

##color(blue)(x) = 9 + 3 = color(blue)(12)##