Before we get to the core of the course, there is a bit of a warm up we’d like to do and introduce some ideas which are useful in the long run. We start with a short review of the math that we need. If you’re comfortable with logarithms, the big-O notation, and the proof by contradiction technique, you can attempt the interactive exercises included within the lessons and then jump straight to the next chapter on graphs.

Note: In general, $\log_b x$ is the inverse of $b^x$, for any positive constant $b$, where $b \neq 1$. This constant $b$ is called the base of the $\log$ function.

The base of a logarithmic function can be any positive number other than 1. Why is that? It has to do with the fact that $1^x$ is always $1$, regardless of the value assigned to $x$. So, this function $1^x$ doesn’t have a well-defined inverse. There are similar reasons for insisting that the base not be a negative number or a zero.

Note: The domain of a logarithmic function is the set of all positive real numbers.

Conventions

Here are the conventions for denoting log functions of base $10$, $2$, and $e$, an irrational number called Euler’s constant.

1. When the base of a log function is not listed, base $10$ is implied. For example, $\log x$ means $\log_{10} x$.

2. When the base is $2$, $\log_2 x$ is written $\lg x$.

3. When the base constant is $e$, $\log_e x$ is written $\ln x$ and referred to as the natural log of $x$.

Let’s check out log functions against different bases.

Notice how there’s a dramatic decrease in the values taken by $\log_{10} n$ compared to the values taken by $\log_2 n$.

This behavior is the opposite of what we would observe for the corresponding exponential functions, where changing the base from $2^n$ to $10^n$ would have a dramatic increase in the values taken by $10^n$.

Properties of log functions

The following four properties hold for a log function, regardless of the choice of base constant $b$. We list these for base $10$ for convenience only:

• $\log xy = \log x + \log y$

• $\log x/y = \log x - \log y$

• $\log x^y = y\log x$

• $\displaystyle\frac{\log_k x}{\log_k y}= \log_y x \qquad \text{for any base } k$

It is also helpful to remember that $\log_b 1 = 0$ for any base $b$.