in this lesson we're going to focus on graphing natural log functions and exponential functions with the base e now the principles learned in sections one and two of this lesson still applies here the way you would graph a regular log is the same way you would graph a natural log and the way you would graph 2 to the x is the same as e to the x but let's work on some examples let's start with the graph e to the x minus two plus three so what we're gonna do is we're gonna set the exponent equal to two things zero and one just as we did before and we're going to make a table so the x values that we're going to choose are if you solve for x it's going to be 2 and 3. the horizontal asymptote is based on this number so y equals three is the horizontal asymptote now we don't need this part anymore let's find out the y value when x is two two minus two is zero e to the zero is one anything raised to the zero power is one so one plus 3 equals 4. so when x is 2 y is equal to 4. now what about when x is equal to 3 what is the value of y so 3 minus two is one e to the one is just e and e is about two point seven one eight but let's round it and say it's two point seven two point seven plus three is about five point seven and now we have enough information to make the graph so let's start with the horizontal asymptote at 3. when x is two y is equal to four when x is 3 y is going to be about 5.7 so very close to 6. the graph is going to start from the asymptote and it's going to follow the two points so the domain is all real numbers from negative infinity to infinity the range is based on the y values the lowest y value is 3 the highest is infinity so the range is from 3 to infinity now let's try an example with the natural log function let's say ln x minus one plus two so this time we're going to set the inside part equal to three things zero 1 and whatever the base is the base of a natural log function is always equal to e and now let's find the value of x so the first one where x equals 1 that is the vertical asymptote and when x minus one is equal to one if you add one to both sides one plus one is two so x is two and if the other one is e plus one where e is two point seven two point seven plus one is about three point seven so now let's find the y values so let's plug into two minus one is one ln one is zero zero plus two is just two now three point seven represents e plus one so if we replace x with e plus one e plus one minus one is basically just e the ones cancel and l and e is always one and one plus two is three so we now have everything we need to graph so let's start with the vertical asymptote at x equals one next we have the point two comma two and then three point comma 3 which is close to 4. the graph is going to start from the vertical asymptote and it's going to follow the two points the lowest y value is negative infinity the highest is infinity so the range therefore is all real numbers negative infinity to infinity now for the domain the lowest x value is one the highest is infinity so the domain is one to infinity so as you can see the techniques used to graph a regular log function is the same as for a natural log function you just have to deal with e and the techniques used to graph let's say 2 to the x is the same for let's say e to the x you