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

*pg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10*

*pg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10*

**2. Logarithms’ definition and basic expressions** **(part 2) **

**2. Logarithms’ definition and basic expressions**

**(part 2)**

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

*Common Logarithm*

**10**

** is written .**

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

*When the number is a multiple of*

*10*we have:

*** for exponential expressions **

*** for logarihmic expressions **

**Examples:**

**Examples:**

**1) the exponential expression written as a logarithmic expression is **

**2) the exponential expression written as a logarithmic expression is **

**3) ** ** the exponential expression written as a logarithmic expression is **

**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*.

*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:*

*Examples:*

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

*832*) that are multiple of

*10*are

*100*and

*1000*.

** is the smallest number and **

** is the largest number and **

**This means that is equal with a number that is between ***2* and *3*.

*2*and

*3*.

**Indeed **

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

*23574*) that are multiple of

*10*are

*10000*and

*100000*.

** is the smallest number and **

** is the largest number and **

**This means that is equal with a number that is between ***4* and *5*.

*4*and

*5*.

**Indeed **

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

*characteristic*and the decimal part is called

*mantissa*.

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

*Example 1*:

*2*is characteristic and

*92*is mantissa.

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

*4*is characteristic and

*372*is mantissa.

*Natural Logarithm* is the logarithm of base . 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 instead of .

*Natural Logarithm*is the logarithm of base . 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 instead of .

** pg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10**

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