Question 723028
Logarithms can have any positive number except 1 as a base. Natural logarithms are just a logarithm with a specific base, the number we call "e". In other words, {{{ln(x) = log(e, (x))}}}.<br>
The graphs of all logarithms with a base which is greater than 1 have similar-looking graphs. Here's the graphs of {{{y = log((x))}}} (whose base is 10) in green, {{{y = log(5, (x))}}} in blue and {{{y = ln(x)}}} in red:
{{{graph(500, 500, -1, 5, -3, 3, ln(x), log(10, x), log(10, x)/log(10, 5))}}}
The main difference in the three is how sharply they curve. The higher the base is the more sharply the graph curves. (Note: The actual graphs do not intersect the y-axis. If it looks like they do, it is just because of a flaw in algebra.com's graphing software.)<br>
For the graph of f(x) = ln(x-e)-1 we just take the graph of y = ln(x) and preform the appropriate transformations. In f(x) we have "x-e" instead of just "x" so the graph of f(x) will be shifted to the right by "e" (which is about 2.8). The "-1" in f(x) will shift the graph down 1. So to sketch f(x), take the graph of ln(x) and shift it to the right by about 2.8 and down by 1.