Logarithms

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What are Logarithms? 

Logarithms are the alternative ways of processing the methods of exponentials. Let's understand logarithms with an example. We know 2 to the power 3 is equal to 8. This is called an exponential equation.

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Let us now assume that you are asked “4 raised to which power is 64?”

The answer replied by you will be 3.

This situation can be explained as a logarithm equation log\[_{4}\] (64) =3

From this, we can say that log base three of twenty-seven is three. 

We can clearly understand from the exponential equation and logarithm equation that there is some relationship between them.

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Define Logarithm

A logarithm is used to raise the power of a number to get a certain number. 

log\[_{2}\] 8 = 3


Logarithm Properties

In logarithm, we will learn about some properties which will help us solve the logarithm equations. We know the logarithm equation has the same relationship with the exponential equation. It also has some similarity between the properties of the logarithm to exponential.

The following are the properties:


Property of Product in Logarithm

It is the sum of the numbers of logarithms

For example,

Solve the logarithm for log\[_{7}\] (3x):

We can see that inside the bracket there are two variables, 3 and x. Now we will use the product rule to solve the logarithm

log\[_{a}\](3x) = log\[_{a}\](3) + log\[_{a}\](x)

We can also simplify two variables of a logarithm into a single logarithm by using the property of the product. Keep in your mind that the bases of logarithm should be the same in two variables to use the property of the product while simplifying the variables into one.

For example,

We cannot use the property of the product to simplify a logarithm which is 

log\[_{4}\](8) + log\[_{6}\](x)


Property of Quotient

It helps you to find the difference between the two variables which are in the form denominator and numerator. Now, let us see an example for our better understanding,

By solving the logarithm of log\[_{5}\] \[\frac{r}{4}\] :

We are going to write this logarithm equation into different forms by using the property of quotients.

log\[_{5}\]\[\frac{r}{4}\] = log\[_{5}\](r) - log\[_{5}\](4)

Now, we will understand how to condense the different forms of a logarithm into dividend and divisor forms. 

log\[_{3}\](4) - log\[_{3}\](h) = log\[_{3}\]\[\frac{4}{h}\]

When we simplify the different form of a logarithm into dividend and divisor form, the bases of logarithm should be the same.

We don’t use the property of quotients when the bases of logarithm are not the same.


Property of Power

This property helps us to know that the power of the log is the exponent times the logarithm of the base of the power.

Let’s see an example,

By solving log\[_{3}\](x\[^{2}\]) :  

Here we will convert the single logarithm into multiple logarithms by using the property of power of logarithm.

log\[_{3}\](x\[^{2}\]) = 2.log\[_{3}\](x) = 2log\[_{3}\](x)

Now we will convert a multiple of a logarithm into a single logarithm.

Let’s take the help of the property of quotient to convert multiple logarithms into a single logarithm

5.log\[_{3}\](9) = log\[_{3}\](9\[^{5}\]) = log\[_{3}\](59049)


The Inverse of Property of Logarithms

From the definition and few examples above, we have understood that the exponential equation and logarithm equation follows the same kind of relationship as function. That means the exponential equation’s inverse is logarithm equations.

When two inverses are written in the form of equations, they equal to y.

So,

f(y)= a\[^{y}\] and g(y)= log\[_{b}\]y

This means: 

f.g= b\[^{log_{b}y}\] = y and g.f = log\[_{b}\] b\[^{y}\] = y 

These are called the inverse properties of Logarithm. 


Application of Logarithms

In this world of modern technology, people are always finding ways to do things in simpler and easier ways. Therefore, people invented calculators and logarithms to make mathematical equations easier to solve. 

So, let us find some more advantages of learning logarithms:

  • Logarithms are used in various fields of science and many other industries.

  • Logarithms help to find the pH value in chemistry because the value for pH can be small, so we use the logarithm to have a range for using it for small numbers.

  • Logarithms are widely used in banking.

  • Logarithms are used to find the half-life of radioactive material.

  • It allows us to find out the earthquake’s intensity.

  • Even in the field of medicine or engineering, we can see some usage of logarithm and its properties.

FAQ (Frequently Asked Questions)

Q1. What are the Different Types of Functions of a Logarithm? 

Ans: Yes, we have two types of logarithm present in logarithm functions and they are as following: 

  • Natural Logarithm 

The natural logarithms are the base of e of the logarithm. We can represent the natural logarithm as log or ln.

For example,

The natural logarithm of 56 can be rewritten as ln 56

The natural logarithm shows us to multiply e to get the desired output.

  • Common Logarithm

The common logarithm is the base 10 of the logarithm. We can represent them as a log or log 10. 

For example,

In logarithm, we can write the 100 as log(100). 

It generally shows the times we need to multiply to get the number 10 as per the requirement.

2. Why is it Necessary to Know the Properties of the Logarithm?

Ans: While studying, you will witness many properties like commutative, associative, and distributive properties. These types of properties are introduced to help us to solve our equations or questions with easier methods.


Similarly, logarithm properties were introduced to help us to solve the difficult and time-consuming equations more simply. To help to solve a difficult equation, we use the three properties of logarithm: property of product of the logarithm, property of quotient of the logarithm and property of power of logarithm.


Hence, we need to know and learn the properties of the logarithm to solve the difficult equations in a simplified manner.