Why does the logarithm require a special notation?

Solution 1:

Suppose you would like to express the fact that, say, $$\lim_{n\to\infty} \left(-\log_e n + \sum_{i=1}^n \frac1i\right) = 0.577\ldots.$$ How do you propose to do this with no $\log$ notation?

Here is another example. Suppose we have a communications channel—say, a telephone cable—over which we can transmit $C$ bits per second.

We want to use this cable to send a sequence of messages, but don't know ahead of time what messages we will need to send (or else there would be no point in sending them!) But suppose we know that each different message $M_i$ will be sent with probability $p_i$. Can we code the messages $M_i$ into bits in such a way that we can send them through this channel?

The answer is that the total information in the message stream, called the entropy of the stream, is

$$E = \sum_i -p_i \log_2(p_i)$$

bits per message, on average, and the rate at which we can expect to send the messages is no more than $C/E$ messages per second, assuming an optimal translation of messages into bits.

How do you propose to express $E$ without using $\log$?

Here is a third example. Let $\pi(n)$ be the number of prime numbers less than $n$, so for example $\pi(10) = 4$, since 2, 3, 5, and 7 are prime. A famous and deep theorem states that:

$$\pi(n) \sim {n \over \ln n}$$

where $\sim$ means that the ratio of the left and right sides approaches 1 as $n$ becomes very large.

How will you state this without using $\log$?

Solution 2:

Let's come up with a very simple problem.

Suppose we wanted to write that $\log_2 8 + \log_3 9 = x$ (here, of course, $x = 5$). What would we write without the logarithm notation? We can't write $2^x + 3^x = 8 + 9$, or things along those lines. A priori, we don't know how much of $x$ comes from the $\log_2 8$ term or the $\log_3 9$ term, so we can't write it as two equations.

It's much harder to write this simple equation without logarithms.

As you learn more math, you'll also learn that it can be helpful to take logs of things in general. Taking logs has the effect of "dropping the exponent", i.e. $\log a^b = b \log a$, and the effect of taking products to sums, i.e. $\log(ab) = \log(a) + \log(b)$. Both of these are very nice things to be able to do, as they can simplify hard problems into simpler problems.

Solution 3:

The concept of the logarithm is not just the notation. The logarithm, i.e.a function $\log:\mathbb{R}_{+}\rightarrow\mathbb{R}$, has many usefull properties. The property that made the logarithm so important is that it turns multiplication into addition:

$$\log(xy)=\log(x)+\log(y).$$

Notice that adding numbers is much simpler than multiplying them. Thus you can multiply various numbers easily if you have a table of logarithms. From a historical perspective that was probably the most important thing that made logarithms so useful and popular among engineers and scientists. So as an important concept it deserves its own symbol: $\log$.