Graph+of+ln+x

im trying not to put in the calc but i cant visualize the graph in order to sketch it... im looking through all my pre calc notes and im i know from one of the units i have to use a parent function, but i dont know what the graph of log(x) looks like...//
 * Question:** //im a little confused on how to graph a log function

I think it was this particular function that caused a couple of the kids from last year to tell me I HAD to give out copies of the packet. Apparently it's one of the things that's left out of PreCalc ... and so I am going to be giving it as a weekend assignment next year. And you will already have done it! How cool is that ....  The first thing we want to talk about is the domain and range. Since the logarithm is an exponent, the range is obviously all real numbers (I can raise 'e' or 10 to any power I want, positive or negative). But what is the domain? The domain is whatever number we get when we put e^(whatever) into the calculator. It can NEVER be negative. It can NEVER be zero. So the domain is all positive real numbers. You may start to see how helpful knowing what the graph of e^x is. In fact y=e^x and y=ln x are inverse functions. If you rotate the piece of graph paper around the line y=x (and look at it from the back) you'll see the graph. The next step is to plot a few points. When the horse and buggy would drop us off at school we would plot maybe 10 or 12 points to get a good idea of what the graph looks like. Most 21st century kids wouldn't think of such a thing .... but try it (there is printable homemade ghetto graph paper on the school website under my name). Here are a few easy points for y=ln x: x y  1 ... 0 because e^0 = 1 e 1 and remember e is about 2.718 .... if you want to graph log x first that's fine e^2 2 so x is a little less than 9. Note that it is OK to use the calculator to figure out what e^2 is .... As long as you don't do it in the graph! e^3 3 and so on, but we also need: e^-1 = 1/e ..... -1 is the y value thats about a third e^-2 = 1/e^2 .... -2 and so on for the negative exponents ... This should help. (Let me know either way ....) it kinda looks like a graph of the square root of x, except moved down so the x intercept is 1 now all i have to do is move it from how the certain problems say to, so like 3ln(4x)+1, the 1 would be the vertical shift. the paper said that "a" (in this case 4) would be the horizontal strech/compressor, but my notes from like november say that the 3 would be "a" and thats confused me a little bit i have this ghetto graph and found a list of values for the ln(x) and i kinda got "ln-happy" and started pluging in random numbers... and it was interesting because ln(-1)= "i" (imaginary square root of -1) times pi... XD//
 * Answer 1:**
 * Follow up Question:** //ok i think i got it ok

Great job!! I'll share something with you ... once you get the idea of plotting numbers, you can do that anytime with a random example like 3 ln (4x) + 1. So ... you can EASILY take the ln of 1 .... the answer is zero. 3*0 + 1 = 1 so that is an easy value to plot. Now ... what makes the INSIDE (the thing we're taking the ln of) equal to 1? Well ... it's when 4x = 1, or in other words when x=1/4. So that's the OLD "x-intercept" only as you noted it's shifted up. So what's the NEW x-intercept? It's when ln (4x) = -1. When is that? When e^(-1) = 4x, or 1/e = 4x, or x=1/(4e) ... about 1/11 or about .09 So ... we have a horizontal compression of a factor of 4, a vertical stretch of a factor of 3, and a vertical shift of +1. I would suggest graphing stuff with the numbers, and then the vertical/horizontal this/that becomes easy .... It's also interesting about ln (-1) = i(pi). At the end of the year we may glimpse what is going on there .... Essentially (BC topic) the value of the function e^x can be expressed as an infinite polynomial series. Once we do that, THEN we can expand the domain of e^x to ANYTHING we can take a power of (like in a polynomial). In other words, we can define e^x for complex numbers .... When we do that we get values that are sometimes complex, and sometimes just negative. And it turns out that, far from being imaginary, complex numbers are THE kinds of numbers that the equations of electromagnetic fields give us. So they represent one aspect of 'reality' better than plain ordinary real numbers do. If you'd rather get an ERROR when you key in ln (-1), change the mode of the calculator from complex to real .... Incidentally, what you stumbled onto is considered by many to be the most beautiful equation in mathematics. It is usually written e^((pi)i) + 1 = 0 because it ties together ALL the special numbers in math in a simple and elegant form. It was discovered a little over 300 years ago by Leonard Euler (pronounced Oiler).... Nice work!
 * Answer/Response 2:**

Note that y=ln x has a vertical asymptote at x=0 ... sqrt(x) does not :-)
 * Supplemental Note:**