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2. Exponents and Logarithms

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When reviewing basic integrals, perhaps you have noticed this somewhat disappointing fact: the formula \[\int x^n dx = \frac{x ^{n+1}}{n+1} + C\] works for all real numbers \(n\) except for \(n = -1\).

Today’s focus will be on what exactly happens when \(n = -1\). This integral turns out to have a name, the “natural logarithm”, and is connected to exponential functions!

Let’s start by defining the function \[f(x) = \int _{1}^{x} \frac{1}{t} dt.\] On a graph, this is the area underneath the graph of \(y(t) = \frac{1}{t}\) between \(1\) and \(x\), positive if \(x > 1\) and negative if \(x < 1\). The domain of \(f\) is \((0,\infty)\) — when \(x \leq 0\), we would be integrating over a huge vertical asymptote, and the area becomes infinitely large and impossible to make sense of.

Please excuse my rather crude artwork.

If we draw a graph of \(f(x)\), we note a few important features:

1. \(f(1) = 0\). This represents the area beneath the graph between \(1\) and \(1\), which is an infinitesimal sliver with no area at all.
2. The area enclosed between \(1\) and \(x\) is always getting larger, but at a slower and slower rate as \(x\to\infty\).
3. This may be unbelievable, and we can’t really prove it, but there is an infinite (negative) area between \(0\) and \(1\).

Together, these three features let us construct the following graph of \(f(x)\):

But there’s more! We observe that \(f\) passes the horizontal line test. It (mysteriously) has a range of \((-\infty, \infty)\), and so its inverse (which we know exists) will have a domain of \((-\infty, \infty)\) and a range of \((0,\infty)\). Here’s what \(f\) and its inverse should look like on the same graph:

You may recognise that \(f ^{-1}(x)\) has a particular shape: it looks like an exponential function! Indeed, it can be shown that \(f ^{-1}(x) = e^x\), where \(e = 2.718281828\ldots\) is known as Euler’s number.

\(f(x)\) is called the natural logarithm, sometimes defined as the inverse of \(e^x\). We often denote \(f(x) = \ln (x)\) instead, and I’ll be using this for the rest of these notes.

You may remember some properties about exponentials, such as \(e^a \cdot e^b = e ^{a+b}\). For instance, \(e^2 \cdot e^3 = e\cdot e\cdot e\cdot e\cdot e = e^5\). Can we expact some relationship involving \(\ln (a)\) and \(\ln (b)\)? It is the inverse of \(e^x\), after all. \[\ln (x) = \int _{1}^{x} \frac{1}{t} dt.\] I’ll suggestively start by writing down \[\ln(ab) = \int _{1}^{ab} \frac{1}{t} dt = \int _{1}^{a} \frac{1}{t} dt + \int _{a}^{ab} \frac{1}{t} dt. \] This is completely unmotivated, but we note that \(\int _{1}^{a} \frac{1}{t} dt = \ln (a)\) by definition! So, \[\ln (ab) = \ln(a) + \int _{a}^{ab} \frac{1}{t} dt.\] The second integral looks pretty similar to our definition of \(\ln\) as well, but it starts at \(a\) instead of starting at \(1\). We can fix this with a \(u\)-substitution: take \(u = \frac{t}{a}\) (so \(t = au\) and \(dt = a\ du\)). Rather than integrating from \(t=a\) to \(t=ab\), we’ll be integrating from \(u=1\) to \(u=b\). Thus, \[\int _{a}^{ab}\frac{1}{t}dt = \int _{1}^{b} \frac{1}{au}a\ du= \int _{1}^{b} \frac{1}{u} du = \ln b.\] Thus, \[\ln (ab) = \ln(a) + \ln (b). \]

In fact, this core identity for the natural logarithm can be stretched a bit to say:

Other Exponentials and Other Logarithms

I’m not sure if we’ll have time to get to this, but I’ll include this for good measure. You may be wondering why \(e\) is so special here; it may seem much more natural to consider \(2^x\) as an exponential function. In a previous remark, we noted that \(b^x\) is a valid exponential function for any real number \(b\neq 1\). For \(b > 1\), this function experiences exponential growth in \(x\). The larger \(b\) is, the more rapid this growth. This creates a spectrum of exponential functions.

Likewise, \(\log_b(x)\) is the inverse to \(b^x\), and the differing shapes of these exponentials also produces a spectrum of logarithms. Here are a few of them on the same graph:

When \(0 < b < 1\), instead \(b^x\) experiences exponential decay, and the shape of \(\log_b(x)\) flips upside-down. Here are a couple examples shown on the same graph, with the logarithms omitted for clarity. I’d encourage you to think about what their graphs would look like, though!

Finally, here are some properties of various exponentials and logarithms that may be useful to know:

From this last property, we have some important “special values” of the logarithms: \(\log_b(b) = \log_b\left( b^1 \right) = 1,\) and likewise \(\log_b(1) = \log_b\left( b^0 \right) = 0.\)