Chapters

## What is a Logarithm?

**mean?**Let’s take a look at the standard

**notation**of a logarithm.

log | Symbol for logarithm |

a | The base |

x | Our input |

y | Our output |

The base of a logarithm **helps** us answer the following: how many of the number a does it take to get x?

Let’s take an **example:**

Here, we have the logarithm of 100 with a base of 10. What the logarithm does is tell us how many **times** we need to multiply the base, 10, to get 100:

As we can see, the answer is **2**. We need to multiply 10 two times to get 100.

## Natural Logarithm

The natural logarithm is a special type of **logarithm.** Take a look at the notation below.

ln | Symbol for natural log |

x | Input |

y | Output |

The natural logarithm as **Euler’s** number as a base. Take a look below.

As you can see, Euler’s number is **approximately** **2.72**. What exactly does this mean? The natural logarithm means that instead of the base taking on any value we want, it is always e.

Take a look at some **examples** below.

ln(5) | |

ln(30) | |

ln(26) |

## What is an Exponent?

An exponent, also known as a power, says how many times a number should be **multiplied** by itself. Take a look at the notation below.

Here are a couple of **examples.**

2x2x2 | |

X multiplied by itself k times | |

4 |

Exponents and logarithms are related in that the logarithm tells us what exponent is needed to get a **certain** value.

Let’s take a look at an **example:**

## Linear Function

In order to understand logarithmic functions, you should be familiar with the concept of functions in general. Functions take a number as **input** and result in another number as an **output.**

The above is an example of a linear function, which is the most basic function in maths. Let’s try plugging in other values into the function above to get **different** outputs.

x | y |

1 | 4 |

2 | 7 |

3 | 10 |

4 | 13 |

5 | 16 |

When we graph these points, we can see that this function creates a **straight line.**

This is why linear functions are called ‘linear,’ because the **relationship** between the input and output of that function is a straight-line relationship.

## Logarithmic Function

Logarithmic functions have the same concept as a linear function: they take in a number as input and result in an output. The **difference,** however, is that the relationship between that input and output is logarithmic instead of linear.

f(x) | Output |

Log with a base of a | |

x | Input |

Let’s take a basic example, with the **common base** of 10.

x | y |

10 | 1.00 |

50 | 1.70 |

100 | 2.00 |

500 | 2.70 |

1000 | 3.00 |

5000 | 3.70 |

10000 | 4.00 |

When we plot this on a graph, we get a curved shape, which indicates a logarithmic **relationship.**

Here are a couple of **rules** for logarithmic functions:

When base a between 0 and 1 | 0 < a < 1 | Slope down |

When base a above 1 | 1 < a | Slope up |

When base a = 1 | a = 1 | Graph undefined |

## Exponential Function

An exponential function is a function that has an exponent in it. Let’s take a look at the **notation.**

While most functions have the value x as a normal number, **exponential functions** have x as a power. Let’s take an example.

x | y |

1 | 3 |

2 | 9 |

3 | 27 |

4 | 81 |

5 | 243 |

6 | 729 |

As you can see, while there is only a difference of 5 between x values 1 and 6, there is a difference of more than 700 in their y values. This is typical **behaviour** of exponential functions, which grow ‘exponentially.’

## Natural Log Function

Natural log functions are exactly like log functions, except that instead of any **base** they have **Euler’s** number as their base. Take a look at the notation.

Let’s take a look at an example of a **natural log function.**

x | y |

1 | 0 |

2 | 0.69 |

4 | 1.39 |

20 | 3.00 |

100 | 4.61 |

150 | 5.01 |

## Example 1

Say that you have the **following** exponential function.

Try this question out on your own **first:** what would the graph look like for this function?

In order to answer, let’s start by plugging in some **numbers** into the function.

x | y |

1 | 1 |

2 | 1 |

10 | 1 |

48 | 1 |

100 | 1 |

1400 | 1 |

As you can see, 1 to the power of any number is just 1. Therefore, the graph is a **horizontal** line at 1.

## Example 2

You are given the following function’s **graph.**

Can you say anything about the a value of the graph just by **looking** at it? To answer this, recall what the a value is.

For exponential functions, there is a **general rule of thumb** when it comes to the a value. Take a look at the rules for the value a for any exponential function.

0 < a < 1 | Slope down |

1 < a | Slope up |

a = 0 | Graph undefined |

So we can tell even before seeing the function that a is between 0 and 1. The function **confirms** this:

The platform that connects tutors and students

Nice lessons like it. My name is David schooling at PNG University of Technology studying Mechanical engineering and your lessons really helped me out in my engineering mathematics. Need more lessons please. Am 21 year old