# Logarithmic Functions

## Definition of Logarithmic Functions
The logarithmic function with base B is defined as the inverse of the exponential function of base B as follows
## Evaluate and Graph Logarithmic Functions
One way to manually evaluate logarithms is to use the relationship between the logarithm and exponential functions given above. The real numbers y, B and x are related by the logarithm and the exponential. The logarithm is the power in the exponential form.
Example 1: Table of values and graphs of logarithmic functions with base greater than 1
Example 2: Table of values and graphs of logarithmic functions with base less than 1
## Properties of the logarithmic functions
From the above values and graphs we conclude the following properties
## Change of base formula
A formula to change a logarithm from any base \( b \) to any other base \( a \) is given by:
\[ \log_b(x) = \dfrac{\log_a(x)}{\log_a(b)} \]
## Explore the more general logarithmic functions using an app
An interactive app is used to explore logarithmic functions and the properties of their graphs such domain, range, x and y intercepts and vertical asymptote.
a, b, c and d are coefficients and B is the base of the logarithm. Logarithmic functions may be explored using an html 5 app shown below.## Interactive Tutorial - Part 1
1 - set a = 1, b = 1, c = 0, d = 0 and B = 2. Check few points on the graph such as \( \log_2 1 = 0 \), \( \log_2 2 = 1 \), \( \log_2 4 = 2 \). Use zoom in and out if necessary.
## Domain and Range of the Logarithmic Function## Interactive Tutorial - Part 2
Let \( f(x) = log_B x \).
## Vertical Asymptote of the Logarithmic Function## Interactive Tutorial - Part 3 \( \log_B 0 \) is undefined. However it is possible to investigate the behavior of the graph the logarithmic function as x gets closer to zero from the right \( x \gt 0 \).
As x gets closer to zero, f(x) decreases without bound. The graph gets closer to the y axis (x = 0). The vertical line x = 0 is called the vertical asymptote. 1 - set a = 1, b = 1, c = 0, d = 0 and change the base. Observe the behavior of the graph close to the y axis.
## Shifting, Scaling and Reflection of the graph of Logarithmic Functions## Interactive Tutorial - Part 4
1 - Investigate base B: set a=1, b=1, c=0 and d=0. Set B to values between 0 and 1 and to values greater than one, take note of the different graphs obtained and explain.
Answers and Solutions to Above Questions in Interactive Tutorial - Part 4.
## More tutorials and self tests on logarithmic functionsnatural logarithmRules of Logarithms and Exponentials - Questions with Solutions. Calculate Exponentials and Logarithms to any Base:. graphing of logarithmic functions. Self Test on solving Logarithmic Equations. Tutorials on Solving Logarithmic Equations. Self Test on Graphing Logarithmic Functions. |