Logarithm as Inverse of Exponential - Maple Help
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Logarithm as Inverse of Exponential

Main Concept

Given  and , with , the logarithm base  of , written  is the exponent to which  needs to be raised to obtain . That is,  means exactly that . Thus, the functions  and  are inverses of each other. The domain of the logarithm base  is all positive numbers. The range of the logarithm base  is all real numbers.

General Logarithms

Recall that the domain and range of an invertible function are just the range and domain of its inverse. Thus, the domain of the logarithm base  function is the range of the  function (all positive numbers) and the range of the logarithm base  function is the domain of the  function (all numbers).

 

Examples: 

• 

 since

• 

 since

• 

 since the logarithmic function  and the exponential function  are inverses of each other.

• 

 for any base , since  for all .

The Natural Logarithm Function

One exponential function is so important in mathematics that it is distinguished by calling it the exponential function. This exponential function is written as  or, particularly when the expression in the exponent is complicated, . The inverse of this function is just as important in mathematics.

 

The Natural Logarithm Function

The natural logarithm function is the inverse of the exponential function, , where   . This function is so important in mathematics, science, and engineering that it is given the name "ln": . Reading out loud, it is pronounced "lawn of x" or often just "lawn x".

 

The graph of the natural logarithm function can be obtained from that of the exponential function by reflection across the line :

 

Exploring the function logb(a) with base greater than 1 and between 0 and 1

Use the sliders below the graphs to change the values of , the base of the logarithmic function  and its corresponding exponential function . For the graph on the left, the base is a number greater than 1. For the graph on the right, the base is a number between 0 and 1. Note that there is no logarithmic function with base . Do you see why not?

 

 

 

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