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.
For each of the following graphs of f \left( x \right) = \log x and g \left( x \right):
Describe the transformation applied to f \left( x \right) to obtain g \left( x \right).
Hence, state the equation of g \left( x \right).
The graph of y = \log_{6} x is transformed to create the graph of y = \log_{6} x + 4. Describe the transformation that could achieve this.
If the function f \left( x \right) = \log_{3} x is translated 5 units to the right, state the equation of the resulting function.
Describe the transformation required to change the graph of g \left( x \right) into f \left( x \right) for each of the following:
g \left( x \right) = \log_{3} x into f \left( x \right) = \log_{3} x + k, where k > 0.
g \left( x \right) = \log_{3} x into f \left( x \right) = \log_{3} x + k, where k < 0.
g \left( x \right) = \log_{10} x into f \left( x \right) = \log_{10} \left(x - h\right), for h > 0.
g \left( x \right) = \log_{10} x into f \left( x \right) = \log_{10} \left(x - h\right), for h < 0.
g \left( x \right) = \log_{10} x into f \left( x \right) = a \log_{10} x, where a > 1.
g \left( x \right) = \log_{2} x into f \left( x \right) = a \log_{2} x, where 0 < a < 1.
Describe the transformation of g \left( x \right) = a \log_{10} x, to obtain f \left( x \right) = - a \log_{10} x.
Consider the graph of y = \log_{6} x which has a vertical asymptote at x = 0. This graph is transformed to give each of the new functions below. State the equation of the vertical asymptote for each new graph:
The graph of y = \log_{4} x has a vertical asymptote at x = 0. By considering the transformations that have taken place, state the equation of the vertical asymptote of the following functions:
y = 2 \log_{4} x - 4
y = 2 \log_{4} x
y = \log_{4} \left(x - 5\right)
y = - \log_{4} x
y = \log_{4} \left(x + 3\right) - 2
Consider the functions f \left( x \right) = \log_{2} x + \log_{2} \left( 3 x - 4\right) and g \left( x \right) = \log_{2} \left( 4 x - 4\right).
Evaluate f \left( 2 \right).
Evaluate g \left( 2 \right).
Is f \left( x \right) = g \left( x \right)?
State the domain for each of the following functions:
y = 5 \log_{5} x - 3
y = \log_{3} \left(x + 5\right) - 4
For any logarithmic function of the form y = a \log_{b} \left(x - h\right) + k, state the range of the function.
A logarithmic function of the form f \left( x \right) = \log_{3} \left(x - h\right) is used to generate the following table of values:
State the exact function used.
Hence, determine the value of f \left( 2.5 \right). Round your answer to three decimal places.
| x | 3 | 5 | 11 | 29 |
|---|---|---|---|---|
| f\left(x\right) | 0 | 1 | 2 | 3 |
Consider the following graph of y = f \left( x \right):
When x = 1, state the value of y.
Describe the transformation that has been performed on the the graph of \\y = \log_{k} x to obtain the graph of f \left( x \right).
Hence, state the equation for f \left( x \right).
Consider the following graph of y=f \left( x \right):
Write down the equation of the vertical asymptote of f \left( x \right).
Describe the transformation that has been performed on the the graph of \\y = \log_{k} x to obtain the graph of f \left( x \right).
Hence, state the equation for f \left( x \right).
The graph of y=f \left( x \right) shown is a transformation of y = \log_{5} x:
Describe the transformation that has been performed on the graph of \\y = \log_{5} x to obtain f \left( x \right).
Hence, state the equation for f \left( x \right).
Consider the following graph of y = f \left( x \right):
When f \left( x \right) = 0, state the value of x.
Describe the transformation that has been performed on the the graph of \\y = \log_{k} x to obtain the graph of f \left( x \right).
Hence, state the equation of f \left( x \right).
The graph of y=f \left( x \right) shown is a transformation of y = \log_{3} x.
Describe the transformation that has been performed on the graph of \\y = \log_{3} x to obtain f \left( x \right).
Hence, state the equation of f \left( x \right).
Find the equation of each of the following functions, given it is of the stated form:
y = k \log_{2} x
y = \log_{4} x + c
Consider the functions graphed below:
Which of these graphs represents a logarithmic function of the form y = -\log_{a} \left(x\right)?
The function graphed has an equation of the form y = k \log_{2} x + c and passes through points A\left(4,11\right) and B\left(8,15\right):
Use the given points to form two equations relating c and k.
Hence, find the values of c and k.
State the equation of the function.
Consider the graphs of the functions
y = \log_{2} x
y = \log_{2} x + 5
y = 5 \log_{2} x
graphed on the same coordinate axes:
Find the value of each function when x = 4:
Match each function to its correct equation:
Describe the relationship between the values of the function f \left( x \right) and the function h \left( x \right), for very large values of x.
Describe the relationship between the values of the function f \left( x \right) and the function g \left( x \right), for very large values of x.
Consider the function f \left( x \right) = \log_{4} \left(k x\right) + 1. Solve for the value of k for which f \left( 4 \right) = 3.
Consider the function f \left( x \right) = \log_{2} 8 x.
Rewrite \log_{2} 8 x as a sum of two terms.
Hence describe how the graph of y = f \left( x \right) can be obtained from the graph of \\g \left( x \right) = \log_{2} x.
Consider the function f \left( x \right) = \log_{3} \left(\dfrac{x}{9}\right).
Rewrite \log_{3} \left(\dfrac{x}{9}\right) as a difference of two terms.
Hence describe how the graph of y = f \left( x \right) can be obtained from the graph of \\g \left( x \right) = \log_{3} x.
Consider the function f \left( x \right) = \log \left( 100 x - 500\right).
Rewrite \log \left( 100 x - 500\right) as a sum of two terms.
Hence describe how the graph of y = f \left( x \right) can be obtained from the graph of \\g \left( x \right) = \log x.
Consider the function f \left(x \right) = \log_{3} x - 1.
Solve for the x-intercept.
Complete the table of values.
State the equation of the vertical asymptote.
Hence sketch the graph of f \left(x \right).
| x | \dfrac{1}{3} | 1 | 3 | 9 |
|---|---|---|---|---|
| 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).
State whether the following features of the graph of f \left(x\right) will remain unchanged after the transformation to g \left(x\right):
The vertical asymptote.
The general shape of the graph.
The x-intercept.
The range.
Sketch the graph of the following functions:
y = \log_{3} x translated 4 units down.
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.
Consider the function f \left( x \right) = \log_{3} \left(x - 4\right).
State the equation of the vertical asymptote of f \left( x \right).
State the coordinates of the x-intercept of the function.
Determine the exact value of f \left( 7 \right).
Sketch a graph of f \left( x \right) = \log_{3} \left(x - 4\right).
Consider the function f \left(x\right) = - \log_{3} x.
Solve for the x-intercept.
Complete the table of values.
State the equation of the vertical asymptote.
Sketch the graph of f \left(x\right) = - \log_{3} x.
| x | \dfrac{1}{3} | 1 | 3 | 9 |
|---|---|---|---|---|
| f \left(x\right) |
Consider the function f \left(x\right) = - 3 \log_{5} x.
Solve for the x-intercept.
Complete the table of values.
State the equation of the vertical asymptote.
Sketch the graph of f \left(x\right) = - 3 \log_{5} x.
| x | \dfrac{1}{5} | 1 | 5 | 25 |
|---|---|---|---|---|
| f \left(x\right) |
For each of the following functions:
Solve for the x-intercept.
Complete the table of values.
State the equation of the vertical asymptote.
Sketch the graph of the function.
| x | \dfrac{1}{2} | 1 | 2 | 4 |
|---|---|---|---|---|
| f \left(x\right) |
f \left(x\right) = 3 \log_{2} x
f \left(x\right) = 3 \log_{2} x - 6
f \left(x\right) = - \log_{2} x + 2
f \left(x\right) = - 2 \log_{2} x + 2
For each of the following functions:
Solve for the x-intercept.
State the equation of the vertical asymptote.
Sketch the graph of the function.
f \left(x\right) = 4 \log_{2} \left(x - 7\right)
f \left( x \right) = - \log_{4} \left(x + 4\right)
f \left(x\right) = \log_{2} \left(x - 1\right) - 4