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 f \left(x\right) = \log_{3} x.
Complete the table of coordinates for the given function.
| Point | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Coordinate | \left(\dfrac{1}{9}, ⬚ \right) | \left(\dfrac{1}{3},⬚ \right) | \left(1, ⬚\right) | \left(3, ⬚\right) | \left(9, ⬚\right) | \left(⬚, 3\right) | \left(⬚, 4\right) | \left(⬚, 5\right) |
Sketch the graph of f\left(x \right), clearly indicating the points C, D and E on the graph.
Consider the function y = \log_{2} x.
Complete the table of values for the function:
Sketch a graph of the function.
State the equation of the vertical asymptote.
| x | \dfrac{1}{2} | 1 | 2 | 4 | 16 |
|---|---|---|---|---|---|
| y |
State whether the following elements are key features of the graph of y = \log_{2} x:
The y-intercept
A vertical asymptote
A horizontal asymptote
The x-intercept
A lower limiting value
An upper limiting value
Consider the function y = \log_{3} x.
Find the x-intercept.
Complete the table of values for \\y = \log_{3} x:
State the equation of the vertical asymptote.
Sketch the graph of y = \log_{3} x.
Is the function increasing or decreasing?
| x | \dfrac{1}{3} | 1 | 3 | 9 |
|---|---|---|---|---|
| y |
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.
Consider the given graph of f \left(x\right) = \log_{5} x:
Determine whether the following statements are true or false.
f \left(x\right) = \log_{5} x has no asymptotes.
f \left(x\right) = \log_{5} x has a vertical asymptote.
f \left(x\right) = \log_{5} x has a horizontal asymptote.
Consider the following function y = \log_{3} x:
State the x-intercept of y = \log_{3} x.
What happens to the value of y = \log_{3} x as x gets larger?
What happens to the value of y = \log_{3} x as x gets smaller, approaching zero?
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 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 graph of y = \log_{5} x:
Graph y = \log_{3} x on the same set of axes.
Consider the graphs of y = \log_{4} x, y = \log_{25} x and y = \log_{100} x graphed on the same set of axes.
Which graph is on the top in the interval \left(1, \infty\right)?
Which graph is on the bottom in the interval \left(1, \infty\right)?
Which graph is on the top in the interval \left(0, 1\right)?
Which graph is on the bottom in the interval \left(0, 1\right)?
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).
The functions y = 3^{x} and y = \log_{3} x have been graphed on the same set of axes:
State the domain of y = 3^{x}.
State the range of y = 3^{x}.
State the domain of y = \log_{3} x.
State the range of y = \log_{3} x.
Describe the relationship between the two functions.
Consider the function y = \log_{2} x.
Complete the table of values for \\y = \log_{2} x:
| x | 1 | 2 | 4 | 8 |
|---|---|---|---|---|
| y |
Hence create a table of values for the inverse function of y = \log_{2} x.
| x | ||||
|---|---|---|---|---|
| y |
Hence sketch the graph of y = \log_{2} x and its inverse function on the same set of coordinate axes, clearly indicating the points found in parts (a) and (b).
Determine the equation of the inverse function of y = \log_{2} x.
Consider the function F \left( x \right) = 4^{x}.
Graph F \left( x \right), the line y=x and the inverse to F \left( x \right) on the same set of axes.
What type of function is the inverse function of F \left( x \right)?
Hence, state the equation of the inverse function.