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 tranformation 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