In the following window we can see the curves of the function y = ln(x) in green, log₂(x) in turquoise, log₃(x) in grey and log₅(x) in red. You can see that the first one is similar to the others.

As we have already mentioned logarithmic functions to base e are of special interest.

Obviously as e>1, the function is of this type (y = ln(x), although it has been represented in the windows in this program as y = log(x). "Do not confuse it with a common logarithm log10(x)").

1.- Note that if the base is greater than 1 the the graph of the function always increases (the smaller the logarithm base is, the "more sharply" it increases).

2.- Note that the Y-axis is a vertical asymptote, so that when x approaches 0 the function tends towards "minus infinity" and when x increases towards infinity, the function does too.

3.- Alter a to whichever value you wish to see the curves of other logarithmic functions. Look, in particular, at the equally commonly used log₁₀(x).

As with similar types of exponential functions, these types of logarithmic functions are of less interest.

The following window select a base of any value where a<1 .

2.- You should also notice that the Y-axis is a vertical asymptote of the curve, this time going upwards. However, when x is very large the function tends to "minus infinity".

As is the case with exponential functions, we sometimes find ourselves with a logarithmic function in which x changes to -x, 2x, x+2, x-1, etc.

In particular ln(x), frequently appears as ln(x+1), ln(x-2), ln(1-x).

Look at how two of them compare to ln(x) in the following window. Change the values of x+1 or x-1 to whichever values you wish to see more examples.

1.- Look at how the curve changes in the window.

2.- Which of the above-mentioned properties change? Go back over the "GENERAL PROPERTIES" SECTION and write down any differences in your exercise book.

3.- When x is changed to (-x) more changes occurs. Why is this? Think about when exactly this function exists now: "the negative values of x now become the values for which the function now exists''

Use the following window, changing the value of log(x) as necessary, remember that it's actually ln(x), to answer the following questions:

1.- What's the difference between ln(x) and ln(-x)?

2.- How does the logarithmic function change when "x" becomes "x+1", "x+2", "x-1", "x-3" and in general to "x ± c"? What about when it changes to 2x, 3x, etc? Look carefully at the domain, the point where it intersects the X-axis and the asymptote.

3.- What happens to the logarithmic function when "-x" changes to "1-x", "2-x" and in general to "c-x"?

4.- Which functions obtained from y = ln(x) onwards would cut the Y-axis at a higher or lower point? Write down some examples.

5.- Write examples of when the following straight lines are vertical asymptotes x = 2, x = -3, x = 1/5, x = c, x = -c.