![](https://i.ibb.co/Lv1gP5Q/cover.png)
## Introduction
Hey it's a me again [@drifter1](https://peakd.com/@drifter1)!
Today's article is another high-school refresher on Mathematics, and more specifically on **Exponents, Roots and Logarithms**.
So, without further ado, let's get straight into it!
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## Exponents
**Exponents**, which are also known as **Powers**, specify how many times a base number should be used in a multiplication where it occurs many times by itself. In other words, if *b* is the **base** number and *p* the power, then the corresponding **multiplication** is:
![](https://quicklatex.com/cache3/56/ql_89498a1f977c575e10fe9d42407f2456_l3.png)
For example:
![](https://quicklatex.com/cache3/d1/ql_d1698f45d40b18e2a711ca1ad0e304d1_l3.png)
When typing on a **keyboard** its common to use the *^* **symbol**. Thus, *a^n* is the same as *a<sup>n</sup>*.
### Negative Exponents
If the exponent is negative then it basically specifies how many times we **divide** by the base number. It might be easier to think of it as fractions of the form 1 by number, which are multiplied by each other.
For example:
![](https://quicklatex.com/cache3/e2/ql_488009c98f10f3eaa6939a1deb56fde2_l3.png)
From these examples we can see that it is also possible to calculate the positive exponent and simply take the **reciprocal** afterwards, as shown below.
![](https://quicklatex.com/cache3/c2/ql_3e62230910f7c0b55dd35ff8d7859cc2_l3.png)
### Exponents and Parentheses
**Parentheses** should be treated with caution (!) as exponents can apply on a single variable or the complete term within parentheses, as shown in the examples below.
![](https://quicklatex.com/cache3/d6/ql_1a4434ad4455a066ee0687cd2d6daad6_l3.png)
The same also applies to **minus** (-) **signs**, as the absence of parentheses affects the outcome of exponentiation:
![](https://quicklatex.com/cache3/e5/ql_8930c92f9c6c4ed8705b8b89c9c86de5_l3.png)
### Exponent of 0 and 1
Of course, an exponent of *1* means the number itself:
![](https://quicklatex.com/cache3/7a/ql_f38aaf789f87d7f4cf0c01c87ed1647a_l3.png)
An exponent of *0* gives a result of *1* for any base except *0*. *0<sup>0</sup>* is **undefined** and requires special handling depending on the problem.
### Square and Cube
The square of a number is defined as multiplying it by itself. In terms of exponents it's thus a power of 2.
![](https://quicklatex.com/cache3/f8/ql_0c9040c40e525b48ec622fd781b9a7f8_l3.png)
Because multiplying two negative numbers leads to a positive number, the square of any number always gives a positive result! The squares of integers are known as perfect squares.
Similarly, if a number is used three times in a multiplication, we are talking about the cube of that number, or a power of 3.
![](https://quicklatex.com/cache3/77/ql_067d6352c71c5b45b24ad9ea13c64f77_l3.png)
When cubing a negative number the result will now be negative. Cubing natural numbers (*>0*) yields perfect cubes.
### Fractional Exponents (and Roots)
Of course, exponents are not limited to integers. They can be **any rational** number, and thus even **fractions**.
In the special case where the exponent is a fraction of the form 1 divided by a natural number, we are talking about a **root**. For example, *1 / 2* specifies a square root (√) and *1 / 3* a cube root.
In general:
![](https://quicklatex.com/cache3/36/ql_1e41a5849ec2d8ff82c63ab360838e36_l3.png)
So, what exactly is a **root**?
The **square root** is the number which when squared gives the number within the root. For example *√9 = 3*, because *3* squared equals *9*. Similarly, the *n*-th root is the number which when raised to the power of *n* yields the number within the root. This is a quite complicated problem! Only perfect squares, cubes etc. are easy to identify or at least simplify.
**Complicated fractions** like *3 / 2* may look complicated by they are not (!), as they can mostly be translated into simple powers and roots, like a cube (*3*) and square root (*1/2*) in that case.
### Exponent Properties
The following properties can be used in order to **simplify expressions** which contain exponents:
![](https://quicklatex.com/cache3/77/ql_9a981151dddef40e76b744084ff48877_l3.png)
So, remember that:
- **Multiplying** two exponents, *m* and *n*, of the same base *x*, is the same as raising *x* to a power which is equal to their **sum** : *m + n*.
- **Dividing** two exponents, *m* and *n*, of the same base *x*, is equal to raising *x* to a power which equals the top exponent **reduced** by the bottom exponent : *m - n*.
- Raising an exponential of a power of *m* by another exponent *n*, is equal to raising directly by a power equal to their **multiplication** : *m x n*.
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## Logarithms
With exponents we specify how many times a number is multiplied by itself. With roots we try to find the number which was raised to that power. So, which question remains now?
Well, having a number which is the result of exponentiation, how many times does a given number need to be multiplied by itself in order to end up with that result? **What exponent needs to be used?** That exactly is a **logarithm**.
For example:
![](https://quicklatex.com/cache3/c4/ql_52340c1dd2c9a18d0497deac700dbdc4_l3.png)
as two *3*'s are needed in order to end up with *9*. The result is basically the required exponent : *3<sup>2</sup> = 9*.
The number which is multiplied is known as the **base** of the logarithm. The previous example had a base of 3.
### Relation of Exponents and Logarithms
It's easy to notice the **relationship** between exponents and logarithms. Mathematically, it can be defined as:
![](https://quicklatex.com/cache3/8a/ql_c467e7a060cbd2676d0ed5ae23e2d88a_l3.png)
### Common Logarithms
Logarithms written without base are known as **common** logarithms, and the base is usually *10*. So, they specify how many times *10* needs to be multiplied in order to get a desired number.
For example:
![](https://quicklatex.com/cache3/8e/ql_d7e406a3354e2d1eed3cbe2fe7f9778e_l3.png)
### Natural Logarithms
Another commonly used base is **Euler's number**, *e*, which is about *2.718*. A logarithm with a base of *e* is known as a **natural** logarithm and denoted with *ln*.
### Base Change
**Changing the base** of a logarithm is as simple as dividing by the logarithm of the previous base with a base equal to the new base:
![](https://quicklatex.com/cache3/6a/ql_c2c4430dc88cec6aad67a07cfd18a76a_l3.png)
Using the same concept it's also possible to **swap** *x* and the base:
![](https://quicklatex.com/cache3/55/ql_d7891a5ba06b36fe1834554a3aa41455_l3.png)
### Logarithm Result
Of course, the result of the logarithm can have **decimals** and be **negative** as well. A negative logarithm basically specifies how many times we divide by the base number. But, be careful! In *log<sub>b</sub>(x)*, *x* must always be **positive**. The logarithm of a negative number is **undefined**!
Also, note that the logarithm of 1 gives us 0 for any base:
![](https://quicklatex.com/cache3/1a/ql_5355fb152201cdeb8ba6e4826eda0e1a_l3.png)
### Logarithm Properties
Similar **properties** to those of the exponents can also be defined for logarithms. Those are:
![](https://quicklatex.com/cache3/73/ql_31a46f56a57e829418e320a607f63c73_l3.png)
And so remember that:
- the logarithm of an **multiplication** is equal to the **sum** of the individual logarithms
- the logarithm of a **division** is equal to the **difference** of the individual logarithms
- the logarithm of n to the **power** of *r* is equal to *r* **times** the logarithm of n
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## RESOURCES:
### References
1. [https://www.mathsisfun.com/algebra/](https://www.mathsisfun.com/algebra/)
2. [https://www.khanacademy.org/math/algebra-home/alg-exp-and-log](https://www.khanacademy.org/math/algebra-home/alg-exp-and-log)
Mathematical equations used in this article, have been generated using [quicklatex](http://quicklatex.com/).
Block diagrams and other visualizations were made using [draw.io](https://app.diagrams.net/).
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## Final words | Next up
And this is actually it for today's post!
I'm not sure about the next article yet, but "All About" is generally targeted towards School Mathematics.
See ya!
![](https://steemitimages.com/0x0/https://media.giphy.com/media/ybITzMzIyabIs/giphy.gif)
Keep on drifting!
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