Properties of Logarithmic Functions

Definition (Logarithmic Functions) For and the logarithmic function has domain base and is defined by The expressions (logarithmic form) and (exponential form) are equivalent. The is called the base in both and and the in is the logarithm and the in is the exponent. Thus a logarithm is an exponent.

Example (Converting Between Exponential and Logarithmic Forms) Convert to exponential form.

    Solution.  We have   

Example (Converting Between Exponential and Logarithmic Forms) Convert to logarithmic form.

    Solution. We have

Definition (Common Logarithmic Function) The logarithmic function with base 10 (common logarithmic function) is usually denoted by instead of .

Definition (Natural Logarithmic Function)  The logarithmic function with base (natural logarithmic function) is usually denoted by instead of .

Example (Graphs of Logarithmic Functions) Using basic transformations of functions we use a basic graph and apply different transformations to obtain the graph of         
    Solution. The graph of can be obtained by graphing and applying the scaling factor of
    

Example (Graphs of Logarithmic Functions) Using basic transformations of functions we use a basic graph and apply different transformations to obtain the graph of     

    Solution. The graph of can be obtained by graphing and applying a horizontal shift left 4 units and then scaling with the factors of and

Example (Graphs of Logarithmic Functions) Using basic transformations of functions we use a basic graph and apply different transformations to obtain the graph of

    Solution. The graph of can be obtained by graphing and then reflecting through the -axis.  

Proposition (Properties of Logarithms) If then

    (i)     for all real numbers         
    
    (ii)
    
    (iii)         
    
    (iv) all real numbers such that

    (v) for all real numbers and with

    (vi)   for all real numbers and with

    (vii) all real numbers such that
    

Example (Properties of Logarithms) Write as the sum or difference of logarithms for

    Solution. We have

Example (Properties of Logarithms) Write as a sum or difference of logarithms for

    Solution. We have

Example (Properties of Logarithms) Write as one logarithm.

    Solution. We have

Example (Properties of Logarithms) Write the expression as the sum or difference of two logarithmic functions containing no exponents for

    Solution. We have

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Cite this as:
Properties Of Logarithmic Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/properties-of-logarithmic-functions.html