Table of Logarithms and Negative Characteristic of Logarithms

As you experienced from the previous article for logarithms, there are situations where we cannot solve equations or simplify expressions by just applying laws of logarithms. It is when having numbers in equations/expressions which cannot be written in terms of a power of certain bases directly. Therefore, there should be another way to continue from that point onward in such situations. That is where the table of logarithms comes into play as mentioned in the previous article and we did not go into deep there. This article is to explain and provide you required knowledge about the table of logarithms, how they can be used, and as a very important topic, negative characteristic of logarithms.

Continue reading to learn about what you would need to know!

What is the table of logarithms?

Rather going through some tradditional approaches to introduce the logarithm table, I prefer to let you all know it in a simple and easily understandable way. For instance, if we calculate 10 to the power 0.6990, we get, 

• 10^0.6990 = 5 (approximately)

Thus, according to the laws logarithms that we studied from previous article, this can be also represented in a way such that,

• lg 5 = 0.6990 

However, if we get a number like 5, 6, or any to find its logarithm with respect to the base 10, we cannot do it just using the mind. As we can see, the answer might be a string of decimals and we cannot find it easily. This was really problematic and to avoid this, the table of logarithms was formed. Actually, the table contains logarithms for the base 10 in an organized manner. Thus, everyone who knows how to use this table can find logarithms for any value in base 10. However, there some important facts to know about the table of logarithms first. 

1. The table is only for finding logarithms for positive values.
   (Logarithm of negative numbers are complex numbers)
2. The table contains logarithms for base 10.
3. All decimal strings inside the table are positive values.
4. It hase been managed to contain numbers with 4 decimal places. 
5. Logarithms of numbers between 0 and 1 are negative numbers.

Apart from those informataion, it should also be noted that log of a number between 0 and 1 results a negative number. This is the most important case where negative characteristics are involved. As well, this might be the most critical part to handle for many learners(students).

Thus, in brief, the table of logarithms is a list of decimal strings for base 10 logarithms listed in an orgernized way. The values were calculated for many times and formed the table for making the future calculations easy by its creators.  

How to prepare to read values in the logarithm table?

Since it contains base 10 logarithms, it is a good practice to represent the required number in scientific way first. Then, logarithm laws make our task easy. For instance, if we take the 1528 as the required number, we can represent it in scientific notation as,

• 1528 = 1.528  X 10³ 

When we apply 'log' for this, we will get

lg 1528 = lg (1.528 x 10³)  

lg 1528 = lg 1.528 + lg 10³

since lg 10³ = 3,

lg 1528 = 3 + lg 1.528

Then, we can get logarithm for 1.528 directly from the logarithm table. 

lg 1.528 = 0.1840 (from the table, approximately)

Therefore, 

lg 1528 = 3 + 0.1840 = 3.1840

Like wise, we can find logarithms for any number using the table. Table 1 shows more examples for finding logarithms below.

Table 1

You can see that the method is straight forward! But why are negative numbers (-1, -2, -3) inside the table under characteristic? That is a good question to raise. Let's discuss about it in following topic.

How does Negative Characteristics of Logarithm Occur?

As previously mentioned that, log of a number between 0 and 1 is negative. You can try to prove it by applying the law we applied above for 0.1528 = 1.52852 X 10^-1. But be careful! Now you need to deal with both a negative characteristic and a positive mantissa at the same time. That is why there is such a notaion for representing them as in the 'Logarithm' column above. 

Representing way in simple terms, 

• Negative characteristic - a number with a bar on top of it
• Positive Manstissa - normal decimal point number

RECALL: Logarithms table only contains positive numbers (mantissas) inside. Thus, when we read value for lg 1.528, it is just 0.1840  (positive). But we have to represent it together with negative characteristic, that is why we need to use this notation to avoid conflicts. 

We are safe in just representing them for one value or separately. However, when there are situations like applying mathematical operations on two or more such representations, it will be diffucult. Let's learn them from the below section.

How to solve problems with negative characteristics?

In such situations as mentioned, we just have to perform a small task that makes our lifes easy. If there are multiple values in calculations, you just have to write them down as two parts (negative & positive part). i.e. as in following Figure 1.

Figure 1

Thus, we can use any technique to calculate such numbers/representatons further. Figure 2 below shows some examples for addition, substraction, multiplication, and division operations. Also, meantime, Figure 3 shows sample calculations/simplifications with them using the values from the table of logarithms. 

Figure 2

Figure 3

Hope this article could give you required knowledge in an understandable way. Let us know about your valuable feedback in the comment section below. You are free to contact us anytime by simply sending a message via email. Use the contact form right there and just send the message. No need to register, pay, or login! If you need more examples or some other specific articles, feel free to let us know. We are happy to wirte for you!

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