Logarithms & antilogarithms , How to find Logs & antilogs using Log Table

So today I'll try to explain some general principles about logarithm & its use, I'll try to keep this guide precise and as clear as possible.

This guide will be especially helpful to students who have recently joined class 11, especially in India.

I'll also give an example to learn the use / application of logarithms, so that you may learn the power of it.

In short, it is everything you need to know about logarithms & antilogarithms (but not as much as wiki!) :P

BASE TO & VALUE

Logarithm is not a very tough thing to understand (like anything else!).

Before learning logarithms we'll need to have information about "Base to" & "Value", terms that I've named. :P lol!

Logarithm & Antilogarithms deal with 3 kinds of numbers, I call them "Base to" & "Value" & "answer". "Base to" refers to the basic sign convention or number used.

"Value" refers to the logarithmic value used for a particular "Base to".

For example,
log_bv = a

Here b= Base To, v= value, a =  answer that we get.

"Base to" remains same mostly, but "value" mostly changes.

The most common "Base To" are:
1. e (exponent), which is like π (pie) but doesn't have a constant value. It varies as e^x where 'x' is any rational number.

2. 10, which is the number 10 we use.

Similar to Logarithms, Antilogarithms also have base & value, they just give the value opposite to Logarithms, i.e. whose log has been calculated (You'll learn calculation later in this article.)

WHAT ARE LOGARITHMS & ANTILOGARITHMS

Photo By ddpavumba / Freedigitalphotos.net

So now that we know "base to" & "value", we can see logarithms.

Take an example, someone asks you to calculate:

Antilog(al)₁₀3, then it'll be equal to:

Antilog(al)₁₀3= 10³

^Similarly,

Antilog(al)_e4= e^4.

^{Now Logarithm is the number associated with the opposite of antilogarithm, where "Base to" remains the same & vice versa. For example,}

Comparing with he above examples,

Logarithm(log)₁₀10³ = 3

&

Logarithm(log)_ee⁴ = 4.

PROPERTIES

So now you'll use the properties of logarithms (logs), to use its power.

Some properties of Logarithms are:

Note: These apply for not only 3 & 4, but for all rational numbers.

1. logarithm(log) (3*4) = Logarithm(log) 3 + Logarithm(log) 4, where "base to" remains the same on both sides of equation & can have any rational value.

2. Log(3/4) = Log 3 - Log 4, where "base to" remains the same on both sides of equation & can have any rational value.

3. Log(3⁴) = 4*Log 3, where "base to" remains the same on both sides of equation & can have any rational value.

Hint: You can use the converse of equations too.
Note: These do not apply antilogarithms.

CALCULATIONS USING LOGARITHMIC TABLE (Log Table)

Now that we know logarithmic properties, well done if you've understood them, so let's get started with the use of log tables.

Log tables use Log ₁₀ v, so I'll not be writing "Base to" here, i.e. they'll give you base to 10 log's answer.

The examples that I'll use below will give you complete clarity about the topic, so they aregolden examples & must be seen.

Calculating logs

Learn one thing that while calculating Logs (not Antilogs!), we use scientific notation, that is decimal is after one number only. But we get answer in non-scientific notation.

Consider:

EXAMPLE 1. Log (0.8372) {Base to 10 remember!}

we'll write it in scientific notation:

= Log (8.372*10<sup>-1).

Now we'll use property of logs.</sup>

= -1 + Log (8.372)

------------------------------------------------------------------------------------------------------------

Now we'll find the Log (8.372), using log table. The procedure will be same for finding values (in table) of antilogs, tangents, sines, etc. The laws/properties will be different though.

In the table, first find 83 (for above example), in first types of column then find the value under 7 for 83 in second kind of columns & then find the value of 2 for 83, in the third kind of columns.

Now add all the 3 values, you'll get required logarithm base to 10.

9227 + 1

log (8.372) = 0.9228

------------------------------------------------------------------------------------------------------------
Now in the above example

log (0.8372) = -1 + Log (8.372)
= -1 + 0.9228
= -0.0772

What if we want the value Log _e  , instead of Log ₁₀  , using Logarithm Table.

So learn a rule,
Log _e = 2.303 Log ₁₀

Calculating Antilogs

Now I'll teach you antilogs, with another golden example.

EXAMPLE 2. Calculate Antilog (86.654), using Anti-logarithmic Table.

Now to do this, 

Antilog (86.654) = 10^86.654

But is it possible to calculate it directly?
No. Of-course not.

Therefore, use this method:

Note: In Antilogs, we use non-scientific notation, but get the answer in scientific notation.

Note: Tables give "Base to" 10 remember!

Al (86.654) = 10⁸⁶ * Al (0.654)

Now use the anti-logarithmic table in the same way as taught in example 1.

The answer is: 4.508 * 10⁸⁶


Calculating Antilogs (Special case)

Did you know that we can also calculate Antilogs of negative numbers, here's another golden example to teach you the same.

EXAMPLE 3. Antilog ( -8.654), using Anti-logarithmic table.

Now for negative numbers, Use the whole number next to the given number, add & subtract it, a mathematical trick.

Al ( -8.654) = Al ( +9 -9 -8.654)

Now +9 -8.654 gives 0.346

= Al (-9 +0.346)

= Al ( 0.346) * 10^-9

Now from Anti-logarithmic table calculate Al (0.346).

Therefore, answer is: 2.218 * 10^-9

SOME IMPORTANT POINTS

So here I have some necessary golden (or silver :P) points, which are actually sometimes confused.
I've also summed up the important points mentioned earlier, so that you may not need to list them separately.


1. Tables ( Both Logs & Antilogs) have base to 10.

2. If we use log_e , it is called natural Log (or ln)

3. Log _e ( ) = 2.303 Log ₁₀ ( ) = Ln ( )

4. Procedure followed to get the values of logs, antilogs, sines, tangents, cotangets, etc. are same in a typical Logarithmic book

5. Logarithms can't be negative, whereas Antilogarithms can have negative values.
And generally, the opposite is true for their answers. Logs can have negative answers & Antilogs can't.

6. While calculating Logs (not Antilogs!), we use scientific notation, that is decimal is after one number only. But we get answer in non-scientific notation.

7. In Antilogs, we use non-scientific notation, but get the answer in scientific notation.

8. Properties of Logs do not apply antilogarithms.

9. General Form of Logs:

log_bv = a

Here b= Base To, v= value, a =  answer that we get.

10. Log 1 with any "Base to" is ZERO 0.

Log_xx = 1, Here x can be 10, e or any other natural number.

PRACTICAL APPLICATIONS

So here I'll give a practical example or application of logarithms, so that you may be able to witness the power of logs.

EXAMPLE 4. Evaluate 245 x 35

Applying logarithms gives
log (245 x 35) = log 245 + log 35
= log (2.45 x 10²) + log (3.5 x 10¹)
= log 2.45 + log 10² + log 3.5 + log 10¹
= 0.3892 + 2 + 0.5441 + 1
= 3.9333

Therefore, 245 x 35= al(3.9333)=8576

Actually LHS 245 x 35= 8575
which is close enough! Isn't it?

And if you have log book with you and no calculator, you won't need to calculate!