Honours GSE Advanced AlgebraName: ______
Unit 5: Exponential and Logarithmic Functions
The Logarithmic Function – Task 7Date: ______Period: _____
So far we have looked at logarithms as exponents, but in this task we will extend our study of logarithms by looking at the logarithmic function. The logarithmic function is defined as for and . Associated with every logarithmic function is a base number. Given an input value, the logarithmic function returns the exponent to which the base number is raised to obtain this input; thus, the output of the logarithmic function is an exponent.
In Task 2 we saw there is a relationship between exponents and logarithms. The ability to go from an exponential expression to a logarithmic expression is powerful. Not surprisingly, there is a connection between an exponential function and a logarithmic function. In fact, implies . Why? The connection between an exponential function and a logarithmic function will be explored more deeply in Unit 6.
For each function given, complete the table of values and then use these points to graph the function on graph paper.
110
20
50
100
1
2
4
8
16
What common characteristics of these functions do you see? In particular, determine the domain and range of the functions and any intercepts. Also describe any characteristics of their graphs such as increasing/decreasing, asymptotes, end-behavior, etc.
Use graphing technology to graph . Does your graph agree with your hand-drawn graph?
Use graphing technology to graph . (Remember you can write any logarithmic expression in terms of common logarithms—this will allow you to graph with your technology.) Does your graph agree with your hand-drawn graph?
Use graphing technology to graph .
How does the graph of the logarithmic function change as the base b changes?
Use graphing technology to graph each function.
How do these graphs compare to the graphs of and and
Use what you know about transformations of functions to explain the relationship between and ?
Does the same relationship hold for the graphs of and ?
For and?
In general, what is the relationship between and ?
Graph . How does this graph compare to that of?
Based on what you know about transformations of functions, describe in words how
transforms the parent function.
Use technology to graph . Key attributes such as domain, asymptote, and intercepts can often be determined algebraically. Consider the following questions to help you determine these attributes algebraically. Confirm your solutions match what you see on the graph.
To determine the domain, solve the inequality . Explain why this makes sense.
To determine the asymptote, solve the equation . Explain why this makes sense.
Use your understanding of the intercepts of any function to determine the intercepts of. HINT: In general, how do you find the intercepts of any function?
.
Use technology to graph . Determine its domain, asymptote, x-intercept, and y-intercept (if applicable) algebraically. Confirm your solutions agree with the graph.
Domain: ______Asymptote: ______
x-intercept: ______y-intercept: ______