What is a Logarithm?

You’ve probably encountered logarithms before, but what do they actually mean? Let’s take a look at the standard notation of a logarithm.

 

log_rules

 

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?

power_explanation

Let’s take an example:

log_10_base

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:

log_explanation

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

 

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Natural Logarithm

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

natural_logarithm_notation

ln Symbol for natural log
x Input
y Output

 

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

eulers_number

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.

log_eulers_base

Take a look at some examples below.

 

ln(5) log_{e}(5)
ln(30) log_{e}(30)
ln(26) log_{e}(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.

power_rule

Here are a couple of examples.

 

2^{3} 2x2x2
x^{k} X multiplied by itself k times
4^{1} 4

 

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

power_log

Let’s take a look at an example:

power_log_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.

output_function

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.

linear_function

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.

log_function

f(x) Output
log_{a} 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.

logarithm_function

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.

functions_power

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

 

    \[ f(x) = 3^{x} \]

 

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.’

positive_logarithm

 

 

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.

natural_log_base

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

 

    \[ f(x) = ln(x) \]

 

x y
1 0
2 0.69
4 1.39
20 3.00
100 4.61
150 5.01

 

natural_logarithm

 

Example 1

Say that you have the following exponential function.

 

    \[ f(x) = 1^{x} \]

 

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.

horizontal_pattern

 

Example 2

You are given the following function’s graph.

decreasing_pattern

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.

power_function

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:

 

    \[ f(x) = 0.4^{x} \]

 

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.