Logarithmic Functions Study Notes

Introduction to Logarithmic Functions

• A logarithmic function can be expressed in the form:
    $f(x) = ext{log}_b(x)$
    where b > 0 and $b <br>eq 1$.

• The function represents the power to which the base, $b$, must be raised to produce the number $x$.

Domain and Range of Logarithmic Functions

1. Domain of $f(x) = ext{log}_b(x)$:
      - The domain is all positive real numbers.
      - Mathematically, this can be expressed as:
        x > 0.

2. Range of $f(x) = ext{log}_b(x)$:
      - The range is all real numbers.
      - Mathematically represented as:
        $y ext{ can take any real value}.$

3. Domain of $f(x) = ext{ln}(x)$:
      - Similar to the logarithmic function, the domain is
        x > 0.
   

4. Range of $f(x) = ext{ln}(x)$:
      - The range is also all real numbers.
      - Thus,
        $y ext{ can take any real value}.$

Graphing Logarithmic Functions

• Functions to Graph:
     - $y = ext{ln}(x)$
     - $y = ext{log}<em>{10}(x)$    - $y = ext{log}</em>{3}(x)$

Similarities Among Graphs:

   - All logarithmic graphs approach the vertical axis (y-axis) but never touch it, indicating that the function's value declines infinitely as x approaches 0 from the positive side.
   - Each log function increases as x increases, indicating a positive slope for all x in the domain.
   

Differences:

   - The rate of increase varies with the base of the logarithm.
   - For example, logarithmic functions with larger bases (e.g., log_{10}(x)) are less steep than those with smaller bases (e.g., log_{3}(x)).

Transformations of Logarithmic Functions

• Given the function $f(x) = ext{log}(a imes x)$ with positive constant $a$, the graph can be transformed by stretching or compressing it horizontally.

Example Discussion:

 - Use technology to sketch the graph of the function for various positive values of $a$:
   - As $a$ increases, the function compresses horizontally, making it steeper.
   - As $a$ decreases, the function stretches, becoming less steep.
   

Logarithm Properties:

• $ext{log}_b(m imes n) = ext{log}_b(m) + ext{log}_b(n)$

• $ext{log}_b\bigg( rac{m}{n}\bigg) = ext{log}_b(m) - ext{log}_b(n)$

• $ext{log}_b(m^k) = k imes ext{log}_b(m)$

Conclusion

• The properties of logarithms provide crucial insights into the behavior of logarithmic functions across different bases. Understanding these behaviors is vital for graphing, transforming, and analyzing logarithmic functions.