Logarithms

Introduction

In mathematics many ideas are related. We saw that addition and subtraction are related and that multiplication and division are related. Similarly, exponentials and logarithms are related.

Logarithms are commonly refered to as logs, are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inverses of exponentials. The logarithm of a number $x$ in the base $a$ is defined as the number $n$ such that $a^n=x$ .

So, if $a^n=x$ , then:

The mathematical symbol for logarithm is ${log}_a\left(x\right)$ and it is read “log to the base $a$ of $x$ ”. For example, ${log}_{10}\left(100\right)$ is “log to the base 10 of 100.”

Logarithm symbols :

Write the following out in words. The first one is done for you.

1. ${log}_2\left(4\right)$ is log to the base 2 of 4
2. ${log}_{10}\left(14\right)$
3. ${log}_{16}\left(4\right)$
4. ${log}_x\left(8\right)$
5. ${log}_y\left(x\right)$

Definition of logarithms

The logarithm of a number is the value to which the base must be raised to give that number i.e. the exponent. From the first example of the activity ${log}_2\left(4\right)$ means the power of 2 that will give 4. As $2^2=4$ , we see that

The exponential-form is then $2^2=4$ and the logarithmic-form is ${log}_{2}4=2$ .

Logarithms

If $a^n=x$ , then: ${log}_a\left(x\right)=n$ , where $a>0$ ; $a\ne 1$ and $x>0$ .

Applying the definition :

Find the value of:

1. ${log}_{7}343$
   $$\begin{array}{c}\hfill \mathrm{Reasoning}:\\ \hfill 7^3=343\\ \hfill \mathrm{therefore},{log}_{7}343=3\end{array}$$
2. ${log}_{2}8$
3. ${log}_4\frac{1}{64}$
4. ${log}_{10}1000$

Logarithm bases

Logarithms, like exponentials, also have a base and ${log}_2\left(2\right)$ is not the same as ${log}_{10}\left(2\right)$ .

We generally use the “common” base, 10, or the natural base, $e$ .

The number $e$ is an irrational number between $2.71$ and $2.72$ . It comes up surprisingly often in Mathematics, but for now suffice it to say that it is one of the two common bases.

Natural logarithm

The natural logarithm (symbol $ln$ ) is widely used in the sciences. The natural logarithm is to the base $e$ which is approximately $2.71828183...$ . $e$ is like $\pi$ and is another example of an irrational number.

While the notation ${log}_{10}\left(x\right)$ and ${log}_e\left(x\right)$ may be used, ${log}_{10}\left(x\right)$ is often written $log\left(x\right)$ in Science and ${log}_e\left(x\right)$ is normally written as $ln\left(x\right)$ in both Science and Mathematics. So, if you see the $log$ symbol without a base, it means ${log}_{10}$ .

It is often necessary or convenient to convert a log from one base to another. An engineer might need an approximate solution to a log in a base for which he does not have a table or calculator function, or it may be algebraically convenient to have two logs in the same base.

Logarithms can be changed from one base to another, by using the change of base formula:

where $b$ is any base you find convenient. Normally $a$ and $b$ are known, therefore ${log}_{b}a$ is normally a known, if irrational, number.

For example, change ${log}_{2}12$ in base 10 is:

Change of base : change the following to the indicated base:

1. ${log}_2\left(4\right)$ to base 8
2. ${log}_{10}\left(14\right)$ to base 2
3. ${log}_{16}\left(4\right)$ to base 10
4. ${log}_x\left(8\right)$ to base $y$
5. ${log}_y\left(x\right)$ to base $x$