Logarithms & Laws of Logarithms with examples

The logarithms are represented with the notation “log” for any base. That means, numbers can be represented in logarithmic way with the required base. Typically, base 10 is the most widely used base in calculations. Apart from that bases like 2, 3, 4, 5, and convenient numbers can be used as required. Hopefully, base 10 logarithm is more familiar to every one since it is most widely used in every school studies and many calculations. 

Why Logarithm is important?

The main reason of using logarithm is to represent large numbers in forms of small numbers as required. Thus, every number is become a lesser value when we write it in logarithms. That is one of the most important usage of logarithm for advanced usage. In addition to that, logarithms can be used for calculations and simplification of mathematical expressions. This is the most important usage of logarithms in school level when solving or simplifying mathematical expressions without calculators.

How to represent numbers in logarithms?

Let's say we have to represent 10,000 using logarithms with base 10, first we need to find the number as a power of the given base. Though this is simple for this example, that might be a critical part in the representation for many students. 

We all know that, 10000 = 10⁴  

Thus, we can represent 10000 using logarithm as,

log₁₀ 10000 = 4  ------ (1)

This is the basic definition for all logarithms. Just have a look at following two lines. They show two main short terms used in logarithms for base 10 and base 'e' for advanced calculations.

• log₁₀ is also represented using short form 'lg'
• log_e is also represented using short form 'ln' - This is mostly applied by students in higher education or advanced level. 

Thus, the above representation (1) can be again represented in the short form as,

log₁₀ 10000 = lg 10000 = 4

Likewise, we can use other bases as well for representations as required. Following are some examples with different bases.

1. 2⁸ = 256, thus log₂ 256 = 8 
2. 5² = 25, thus log₅ 25 = 2
3. 13⁰ = 1, thus log₁₃ 1 = 0
4. 10⁰ = 1, thus lg 1 = 0
5. 15¹ = 15 , thus log₁₅ 15 = 1  
   

The representation is just a simple approach to understand and have the power to solve different complex mathematical expressions. However, there is no usage of the representation of logarithms alone. That is where laws of logarithms come into play! Representations with laws are more powerful and can be used anywhere as needed. Figure 1 shows the laws of logarithms that can be used in anywhere as below. 

Figure 1

You may have heard about the table of logarithms for calculations based on logarithms. In simple terms, there is no use of knowing the table of logarithms if we do not know how to apply laws correctly! Thus, the better practice of applying laws are really really important. There are many sets of equations and expressions that can be solved without using the table but just using laws of logarithms.Therefore, it is good to have a proper practice of applying laws in mathematical expressions first. 

Try to solve following equations by just using the laws mentioned in the figure 1.

1. lg 16 + lg x = lg 8 + lg 2 
2. 3 lg x + lg 96 = 2 lg 9 + lg 4
3. 3 lg x + 2 lg 8 = lg 48 + (1/2) lg 25 – lg 30
4. x = log₃ 27 + 2 log₃ 3 – log₃ 3
5. x = lg 26 + lg 119 - lg 51 - lg 91

Have you faced any trouble? You may! Though the laws are mentioned above, there is no way describing their usage. Thus, you have to know how to apply those in actual calculations. To solve equations like above, we need to know some basic norms as follows. Have a look at following figure 2 to know some properties of logarithms for applying in equations.

Figure 2

No matter still you cannot solve them. Figure 3 shows the way to solve them in the best way. Follow the instructions and see how they have been solved!

Figure 3

Look at the question number (5) above. The answer cannot be found by just applying laws! It is because the number inside the 'lg' function cannot be represented as a power (integer as a power) of base 10. In such situations, we cannot get the exact answer. Therefore, we need another way to solve such problems. That is where the table of logarithms comes to play! What is the table of logarithms and how to use it to do such calculations?

This article provided you a better explanation about basic laws and how to apply them for mathematical calculations in a proper way. The next best usage of logarithms are the use of Table of Logarithms. You can study them from our NEXT ARTICLE here! You may have some doubts about the given details. Thus, do not hesitate to start conversation in comment section below and ask anything by contacting us! You can contact us via the contact form right there. If you subscribe WitCentre, you can get our new articles to your finger tips. 

Thanks for reading!