1. Exponential Functions and Logarithmic Functions
1. Exponential Functions
1. The function (where and ) is called an exponential function with base .
2. Properties of the exponential function (where and ):
1. The domain is the set of all real numbers, and the range is the set of all positive real numbers.
2. When , as the value of increases, the value of also increases.
   When , as the value of increases, the value of decreases.
3. The graph passes through the point and has the -axis as an asymptote.
2. Logarithmic Functions
1. The function (where and ) is called a logarithmic function with base .
2. Properties of the logarithmic function (where and ):
1. The domain is the set of all positive real numbers, and the range is the set of all real numbers.
2. When , as the value of increases, the value of also increases.
   When , as the value of increases, the value of decreases.
3. The graph passes through the point and has the -axis as an asymptote.


2. Applications of Exponential and Logarithmic Functions
1. Exponential Functions
1. When and ,
   
2. When ,
   
   When ,
   
2. Logarithmic Functions
   For and :
1. When and ,
   
   
2. When ,
   
   When ,