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As we mentioned above the relationship between exponents and logarithms is \(x=e^y \to \ln(x) = y\). There are a couple of ways to think about this. First, they are the same equation in that these are just two different ways to write the same thing. Another way to think about it is that the functions (yes, logarithms are functions that pass the vertical line test) \(f(x) = e^x\) and \(g(x) = \ln(x) \) are inverses of each other. You can see that in this plot.

The top (blue) graph is \(f(x)=e^x\) and the lower (green) graph is \(g(x)=\ln(x)\). We have also included the line \(y=x\) for reference to help you see the inverse relationship. [If you haven't learned about or don't remember inverse functions, don't worry. You can go to the inverse functions page to learn everything you need to know about inverse functions for calculus.]

You can see from the graph that only positive non-zero values of \(x\) are allowed as input for the logarithm function. However, you can get any real value out of the function. There is a vertical asymptote at \(x=0\) and, although it looks like it is leveling out as \(x\) goes off to infinity, that is not the case. It just keeps going up and up infinitely.

Although this graph is actually of \(g(x)=\ln(x)\), the general shape of the graph is the same for all logarithm functions, including the fact that the point \((0,1)\) is on all graphs. Now let's look at how to actually graph logarithm functions.

Here is a short video discussing how to graph logarithm functions using an example. It is important to have an idea what a logarithm graph looks like. You will need to know this when working with continuity in calculus.