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Math

Attribution

Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in October 2020. The graphs are generated using Desmos. This work is licensed under a Creative Commons BY 4.0 International License

Defining Logarithms

 

The logarithmic function with base b is defined as:

                                                                                 

where x is called the argument of the logarithm and y is the value of the logarithm. The base of the logarithm, b, is always a positive number. One can use a calculator to find the value of a logarithmic function.

 

Example: What is the value of ?
The base of the logarithm is 9 and the argument is 4. We can use a calculator to find the value of this logarithm:

                                                                       

 

Properties of Logarithms.

1)The logarithm of a product is the sum of the logarithms:
                                                                 

 

2) The logarithm of a division is the difference of the logarithms:

                                                                         

 

3) The logarithm of a power:
                                                                               

 

4) For any base b, 
                                                                                       

 

5)  For any base b, 
                                                                                          

 

Example: Use the logarithmic properties to simplify 

 

We simplify this  logarithm using property 2):

 

Example: Use the logarithmic properties to simplify 

 

Step 1) We use property 1) to combine the first two logarithms:

Step 2) We use property 2) to combine the two logarithms

Step 3) Using property 4) we get:

 

Example: Use the logarithmic properties to simplify

 

Step 1) We use  and then property 3):

Step 2) We use  and then property 3):

Step 3) Finally, using property 4) we get:

 

 

 

 

 

 

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