3.09 Graphs of logarithmic functions
Consider the functions graphed below:
Which of these graphs represents a logarithmic function of the form y = \log_{a} \left(x\right)?
Consider the function y = \log_{2} x.
Complete the following table of values:
| x | \dfrac{1}{2} | 1 | 2 | 4 | 16 |
|---|---|---|---|---|---|
| y |
Sketch a graph of the function.
State the equation of the vertical asymptote.
Consider the function y = \log_{4} x and its given graph:
Complete the following table of values:
| x | \dfrac{1}{16} | \dfrac{1}{4} | 4 | 16 | 256 |
|---|---|---|---|---|---|
| y |
Find the x-intercept.
How many y-intercepts does the function have?
Find the x-value for which \log_{4} x = 1.
Sketch the graph of y = \log_{5} x.
Consider the function y = \log_{4} x.
Complete the table of values.
| x | \dfrac{1}{1024} | \dfrac{1}{4} | 1 | 4 | 16 | 256 |
|---|---|---|---|---|---|---|
| y |
Is \log_{4} x an increasing or decreasing function?
Describe the behaviour of \log_{4} x as x approaches 0.
State the value of y when x = 0.
Consider the function y = \log_{a} x, where a is a value greater than 1.
For which of the following values of x will \log_{a} x be negative?
x = - 9
x = \dfrac{1}{9}
x = 9
\log_{a} x is never negative
For which of the following values of x will \log_{a} x be positive?
x = 5
x = - 5
x = \dfrac{1}{5}
\log_{a} x will never be positive
Is there a value that \log_{a} x will always be greater than?
Is there a value that \log_{a} x will always be less than?
Consider the given graph of the logarithmic function y = \log_{a} x:
Is \log_{a} x an increasing or decreasing function?
Which is a possible value for a,\dfrac{2}{3} or \dfrac{3}{2} ?
Consider the functions y = \log_{2} x and y = \log_{3} x.
Sketch the two functions on the same set of axes.
Describe how the size of the base relates to the steepness of the graph.
Consider the given graph of f \left( x \right) = \log_{k} x:
Determine the value of the base k.
Hence, state the equation of f \left( x \right).
Consider the functions f\left(x\right) = \log_{2} x and g\left(x\right) = \log_{2} x + 2.
Complete the table of values below:
| x | \dfrac{1}{2} | 1 | 2 | 4 | 8 |
|---|---|---|---|---|---|
| f\left(x\right)=\log_2 x | |||||
| g\left(x\right)=\log_2 x + 2 |
Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.
Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).
Determine whether each of the following features of the graph will remain unchanged after the given transformation:
The vertical asymptote.
The general shape of the graph.
The x-intercept.
The range.
Consider the functions f\left(x\right) = \log_{2} \left( - x \right) and g\left(x\right) = \log_{2} \left( - x \right) - 3.
Complete the table of values below:
| x | -8 | -4 | -2 | -1 | -\dfrac{1}{2} |
|---|---|---|---|---|---|
| f\left(x\right)=\log_2 \left( - x \right) | |||||
| g\left(x\right)=\log_2 \left( - x \right) - 3 |
Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.
Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).
Determine whether each of the following features of the graph will remain unchanged after the given transformation:
The vertical asymptote.
The general shape of the graph.
The x-intercept.
The domain.
Sketch the graph of the following functions:
y = \log_{3} x translated 4 units down.
y= \log_{2} x + 4.
The graph of y = \log_{6} x is transformed to create the graph of y = \log_{6} x + 4. Describe a tranformation that could achieve this.
For each of the following functions:
State the equation of the function after it has been translated.
Sketch the translated graph.
y = \log_{5} x translated downwards by 2 units.
y = \log_{3} \left( - x \right) translated upwards by 2 units.