# What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something that exponents already do just fine.

• logs "undo"/"invert" what exponential functions do. Aug 2, 2014 at 21:34
• Why invent subtraction to do something that addition already does fine?
– user14972
Aug 2, 2014 at 21:42
• It is rumored that Old MacDonald suggested its use: $$e-i-e-i-\text{little o}$$ Aug 2, 2014 at 21:48
• @ozo, whoever told you that mathematicians don't really care for applications lied to you. Moreover, we can define lots of things that we do not study... Aug 2, 2014 at 23:36

There's a lot of uses for logarithms

There are things you can solve without logarithms. Take for example:

$$4.9^x=34.185$$

Without logarithms you might try taking the $x$'th root, and really just end up going in circles, with logarithms, we can solve this with logarithms this way.

$$\log 4.9^x=\log34.185$$ $$x\log 4.9=\log34.185$$ $$x=\frac{\log34.185}{\log 4.9}$$

Given any $b > 0$ then $f: \mathbb{R} \to (0, \infty)$ by $f(x) = b^x$ is a bijective, that is one-to-one and onto, function. This means $f^{-1}:(0, \infty) \to \mathbb{R}$ exists. Further more we denote this inverse function by $f^{-1}(x) = \log_b(x)$. That is $\log_b$ is exactly the inverse function to the exponential function with base $b$.

So, why do logarithmic functions exist? It is because exponential functions are one-to-one. Whether you want to use logarithms or not they exist because a bijective function has an inverse.

Logarithms put numbers on a human-friendly scale, such as the Richter scale, the Decibel scale, etc... For example consider the following data, and see how it is easier when dealing with logarithms to compare the different distances:

$$\begin{array}{|l|c|c|} \hline & \text{Distance }x & \log_{10}(x) \\ \hline \text{Earth to Alpha Centauri} & 4.1 × 10^{16}\,\rm m & 16.61 \\ \hline \text{Earth to Pluto} & 5.9 × 10^{12}\,\rm m & 12.77 \\ \hline \text{Toronto to Vancouver} & 3.4 × 10^6\,\rm m & 6.53 \\ \hline \text{Height of average adult male} & 1.78\,\rm m & 0.25 \\ \hline \text{Thickness of a human hair} & 1.0 × 10^{-4}\,\rm m & –4.00 \\ \hline \text{Bohr radius} & 5.0 × 10^{-11}\,\rm m & –10.3 \\ \hline \end{array}$$

See: Logarithmic scale.

But if logarithms have lost their role as the centerpiece of computational mathematics, the logarithmic function remains central to almost every branch of mathematics, pure or applied. It shows up in a host of applications, ranging from physics and chemistry to biology, psychology, art, and music.
— Eli Maor in e: The Story of a Number

• The decibel scale is logarithmic precisely because we hear volume logarithmically. To give another example, we hear pitch logarithmically as well. Aug 2, 2014 at 23:38

Logarithms were originally invented to make multiplication (as in, actually computing the product of two numbers by hand) easier. They were developed by one man, John Napier, in the 16th century, specifically as a method for doing by-hand multiplication.

The reasoning was as follows. It had already been noticed at the time that if you consider the sequence of powers of a number, say $1, 2, 4, 8, 16...$, then multiplying any two numbers in that sequence can be done simply by adding together their positions in the sequence. This is simply another way of saying $x^ax^b=x^{a+b}$. Unfortunately, most numbers are not in the sequence $1, 2, 4, 8...$, so the trick isn't very useful.

Napier's original idea was to simply use a very small base, say $1.000001$. Then the sequence of its powers will grow very slowly, and so even though not every number is in the sequence, every number will at least be close to an element of the sequence. So if you need to multiply together two numbers, you can just approximate them by elements of the sequence and then use the trick discussed in the previous paragraph.

Using the concept of fractional exponents, which I'm sure you've seen if you're studying logarithms, we can do away with having to use a small base by effectively filling in the gaps between the elements of the sequence.

That's why logarithms were originally investigated. Nowadays we don't need to resort to tricks like that just to do multiplication. But once they had a name, they started showing up in all kinds of places. This is why we teach students about logarithms today. For example, in order to integrate $\frac 1 x$ in calculus, you "need the logarithm". Of course, you could just numerically integrate it, but it's useful to know that the result of that integration is actully a function with certain algebraic properties and which turns up as the answer to other problems as well.

Why invent logarithms to do something that exponents already do just fine.

Exponents do the exact opposite of logarithms, so this question is a non-sequitur. However, if you are inquiring about practical uses of logs, they can range from banking (interest rates, for instance) to computing the age of the Earth (based on the half-lives of various radioactive elements), biology (since our sense organs perceive the surrounding world in a logarithmic manner), to engineering (the so-called logarithmic scale, which makes the study of many processes so much easier), etc. I would also suggest reading the following section.