Logarithm Table

Created by Paul Tauriki on 02-07-08.

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Steps taken to create Logarithm Table.pdf: Created Logarithm Table in MS Exel 2003 Exported exel file to MS Word 2003 Opened MS Word file in Open Write and saved as Open Write file Exported Open Write file to PDF file (Lossless)

The following is a Logarithm Table with values rounded to three significant figures for numbers between 1 and 10. In order to use it for numbers less than one and greater than ten, the numbers have to be rounded first to three significant figures then converted to Standard Form before reading the logarithm values from the table.

View Logarithm Table in PDF format

Instructions on How to use the Logrithm Table

This is a technique to simplify harder Maths operations such as multiplications and divisions. Instead of doing multiplication we will do the addition and instead of doing division we will do the subtraction.

Adding instead of Multiplying

Using the Indice Identity where

[math]a^m \times a^n = a^{m + n}[/math] (Note that the bases are the same)

Similarly

[math]10^m \times 10^n = 10^{m + n}[/math]

For example

[math]23.9 \times 567[/math]

First of all we need to convert the above to Standard Form, which is

[math](2.39 \times 10^1) \times (5.67 \times 10^2)[/math]

Now if we look up in the logrithm table for 2.39 we will find 0.378 and looking up 5.67 gives us 0.754. These means

[math]2.39 = 10^{0.378}[/math] and

[math]5.67 = 10^{0.754}[/math].

Replacing 2.39 with [math]10^{0.378}[/math] and 5.67 with [math]10^{0.754}[/math] in the above and discarding the brackets, we will have

[math]10^{0.378} \times 10^1 \times 10^{0.754} \times 10^2[/math]

[math]= 10^{0.378 + 1 + 0.754 + 2}[/math]

[math]= 10^{4.132}[/math]

This can be written as

[math]10^{0.132} \times 10^4[/math].

We need to convert back [math]10^{0.132}[/math] reading the table backward. Reading 0.132 from the table but reading it backward i.e find 0.132 in the body of the table and read the number (from right and top) gives us 1.355, since 1.32 lies between 1.30 and 1.34. Replacing [math]10^{0.132}[/math] with 1.355 in the above gives us

[math]1.355 \times 10^4[/math]. This is the same as 13550. Since the table values are rounded to three significant numbers so our answer should be 13600 (3 s.f.).

Subtracting instead of Dividing

Recall another Indice Identity which is:

[math]\frac{a^m}{a^n} =a^{m - n}[/math], for a ≠ 0

Similarly we can have

[math]\frac{10^m}{10^n} =10^{m - n}[/math]

For example

[math]\frac{9780}{450}[/math]

Converting the above to Standard Form gives us

[math]\frac{9.78 \times 10^3}{4.5 \times 10^2} = \frac{9.78}{4.5} \times 10^1[/math]

Looking up the Logarithm Table for 9.78 gives us 0.990 and 4.5 gives us 0.653. This means that 9.78 = [math]10^{0.990}[/math] and 4.5 = [math]10^{0.653}[/math]. Replacing into the above gives us

[math]\frac{10^{0.990}}{10^{0.653}} \times 10^1 = 10^{0.990 - 0.653 + 1} = 10^{1.337}[/math]

This can be written as

[math]10^{0.337} \times 10^1[/math]

Now we look up 0.337 in the table but reading the table backwards gives us 2.175 since 0.337 is between 0.336 and 0.338. Substituting into the above gives us

[math]2.175 \times 10^1[/math]. So our answer is 21.75 but when rounded to 3 significant figures we get 21.8 (3 s.f.).