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Logarithms – definition and basic expressions (part 2) – pg.3/10

Posted in Math Theory by Lia on 04/08/2013

pg. 1, 2, 3, 4, 5678910

 

2. Logarithms’ definition and basic expressions (part 2

Common Logarithm is the logarithm of base 10. Common logarithms are written without the base.

{log_{10}} p is written { log p}.

 

a) When the number is a multiple of 10 we have:

*   for exponential expressions {10^n} = {underline {10...........................0}under{n = number of  zeros that follows 1}}

*   for logarihmic expressions {log}{underline {10...........................0}under{n = number of  zeros that follows 1}} = n

 

Examples:

1)    the  exponential expression {10^0} = 1 written as a logarithmic expression is {log}1 = 0

2)    the  exponential expression {10^3} = 1000 written as a logarithmic expression is {log}1000 = 3

3)     the exponential expression {10^4} = 10000 written as a logarithmic expression is {log}10000 = 4

 

b) When the number is not a multiple of 10 we consider the two closest numbers that are multiple of 10 one smaller and one larger than our number. The logarithm for our number should have a value in between the logarithm’s values for the two closest numbers that are multiple of 10.

 

Examples:

1)    Let’s consider {log}832. The closest numbers with our number (832) that are multiple of 10 are 100 and 1000.

{{}{100}{}} is the smallest number and  {log}100 = 2

{{}{1000}{}} is the largest number and  {log}1000 = 3

This means that {log}832 is equal with a number that is between 2 and 3.

Indeed {log}832 {approx} 2.92

 

2)    Let’s consider {log}23574 The closest numbers with our number ( 23574) that are multiple of 10 are 10000 and 100000.

{{}{10000}{}} is the smallest number and  {log}10000 = 4

{{}{100000}{}} is the largest number and  {log}100000 = 5

This means that {log}23574 is equal with a number that is between 4 and 5.

Indeed {log}23574 {approx} 4.372

 

The integer part of a common logarithm result is named characteristic and the decimal part is called mantissa.

For the result in Example 1:  2 is characteristic and 92 is mantissa.

For the result in Example 2:  4 is characteristic and 372 is mantissa.

 

Natural Logarithm is the logarithm of base e{}(e {approx} 2.7128). This logarithm is very useful in some practical Advanced Math. Some times this logarithm is called Napierian Logarithm (for John Napier, a Scottish mathematician, the inventor of logarithms). This logarithm is written {{}{ln}{p}} instead of {{}{log_e}{p}}.

 

 pg. 1, 2, 3, 45678910

 

 

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