Logarithm Table

Have you ever heard about earthquake magnitudes? Did you know that the world's largest earthquake occurred in the year 1960 in a place near Valdivia in Chile? It was reported to be magnitude 9.5 by the United States Geological Survey.

A large earthquake is actually meant to be millions of times bigger than a minor one. If you would have tried to create a bar graph in which the bars have sizes 10, 100, and 10,00,000 it would be something that would draw you crazy. The bars that correspond to the sizes 10 and 100 would be too small to be visible, and you would not be able to communicate through those graphs that one of them is just ten times larger than the other. But, if you would choose to take the logarithm of each number, you would get 1, 2, and 5. This would make your bar graph communicate the right information appropriately. With such a handy tool called logarithms, we can precisely estimate about an earthquake that is reported with a magnitude of 8.0 to be ten times bigger than a 7.0 earthquake. Similarly, a 7.0 earthquake is also ten times bigger than a 6.0 earthquake and so on. 

Now that we have come to know where logarithms can be used. Next, we need to understand what actually is the logarithm. Logarithms help you find the most likely cause for an effect.  Assume, there is a raise from ﹩100 to ﹩150 in 5 years. With just this much of data, using logarithms, we can find a possible cause. A continuous return of 8.1% would be the reason for the change. The value 8.1 is obtained by finding the value \[\frac{ln(\frac{150}{100})}{5}\]. This might not be the actual cause of that change in 5 years. However, it is a smooth average that we have obtained to compare with other changes. \[\frac{ln(\frac{150}{100})}{5}\]. In short, logarithms can be used to estimate how long will it take for something to grow or decay exponentially. Some examples are money growing with a fixed interest rate, the time for a certain increase of bacteria in a dish, the sound produced by a bell, radioactive decay and many more. 

There is another notable purpose of using logarithms. The problem faced during the task of working out especially arithmetic operations using very large numbers was the actual cause for logarithms to have come into existence. Logarithms have the ability to put every number on a human-friendly scale. Logarithms help to shrink the numbers of very high magnitude to a smaller one which our brains can deal with easily.

Let us learn more about logarithmic functions.


Logarithmic Function Definition

We can define logarithm as follows:

Y is said to be the logarithm of a number X if 10 (or any other base) raised to Y gives X.

 Y = logₐX is the same as X = ay


An exponential function can be expressed in logarithmic form. In the same way, all logarithmic functions can also be in their respective exponential forms.

In both forms, there are two restrictions which are X > 0 and 3.162 X 10¹.


How to Use the Log Table?

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To determine the logarithm of a number, we should use the logarithm table.  Let us walk through the steps involved in finding a logarithm.

Step 1: Pick the Right Table

To find the value of logₐX, you have to pick the base -‘a’ table. If a=10, then the log table to use is the base-10 table.

Step 2: Write the number in scientific notation. For example, 31.62 is written as 3.162 X 10¹. So now the power of 10 is 1. So 1 is the characteristic part of the resulting logarithmic value.

Step 3: Look for the cell that is at the intersecting point where the corresponding row is labelled with the first two digits of the number and the corresponding column header with the third digit of a number. Hence for the number 31.62, ignore the decimal point and look at the cell at row 31 and column 6. The value would be 0.4997. 

Step 4: As we have one more digit in the number 31.62, we must stay in the same row and find the value of cell at the MEAN DIFFERENCE column number 2 as it is the fourth digit of the 31.62. The value is 3.  Now add this to previous value , 0.4997 to get 0.4997 + 3 = 0.5000. This is called the mantissa part.

Now join both the characteristic and mantissa part to get the logarithm of the number .

So log₁₀31.62 = 1 + 0.5000 = 1.5000

The Characteristic is the integer part of the logarithmic form of the number and Mantissa is the fractional part of the logarithmic form of that number.


Question:

Find the value of log 287.2

Solution:

Step 1: Characteristic Part is found out by first converting the given number to its scientific notation.  28.72 X 10². So the characteristic part is 2.

Step 2: Go to the cell which is at row number 28 and  at column number 7. So the value corresponding to that row and column is 4579.

Step 3: Go to the cell which is at row number 28 and at the mean difference column 2. The value corresponding to that row and  mean difference column is  3

Step 4: Add the result which is obtained in step 2 and 3. The resulting number is 4582. This is termed as the mantissa part.

Step 5: Finally we combine the characteristic part which is obtained in Step 1 and the mantissa part obtained in step 4. So, it becomes 2.4582.

Therefore the value of log 287.2 is 2.4582.


Question:

Find the value of log 0.34

Solution:

Step 1: Characteristic Part is found out by first converting the given number to its scientific notation. 3.4 X 10⁻¹. So the characteristic part is -1.

Step 2: Go to the cell which is at row number 34 and at column number 0. So the value corresponding to that row and column is 5315. This is termed as the mantissa part.

Step 3: Finally we combine the characteristic part which is obtained in Step 1 and the mantissa part obtained in step 4. So, it becomes 0.5315-1=-0.4685

Therefore the value of log 287.2 is -0.4685.