5.06 Logarithmic graphs and scales
For each of the following functions:
Complete the following table of values:
| x | 0.3 | 1 | 2 | 3 | 4 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|---|---|
| y |
Sketch the graph of the function on a number plane.
Consider the function y = \log_{3} x.
Complete the following table:
Sketch the graph of the function on a number plane.
| Point | Coordinates |
|---|---|
| A | \left(\dfrac{1}{9},⬚\right) |
| B | \left(\dfrac{1}{3}, ⬚\right) |
| C | \left(1, ⬚\right) |
| D | \left(3, ⬚\right) |
| E | \left(9, ⬚\right) |
| F | \left(⬚, 3\right) |
| G | \left(⬚, 4\right) |
| H | \left(⬚, 5\right) |
Consider the function y = \log_{2} x.
Find the x-value of the x-intercept.
Complete the following table of values:
| x | \dfrac{1}{2} | 1 | 2 | 4 |
|---|---|---|---|---|
| y |
State the equation of the vertical asymptote.
Is \log_{2} x an increasing or decreasing function?
Consider the function y = \log_{\frac{1}{5}} x.
Find the x-value of the x-intercept.
Complete the table of values:
| x | \dfrac{1}{25} | \dfrac{1}{5} | 1 | 5 | 25 |
|---|---|---|---|---|---|
| y |
State the equation of the vertical asymptote.
Sketch the graph of the function on a number plane.
Is \log_{\frac{1}{5}} x an increasing or decreasing function?
Consider the function y = \log_{3} x.
Complete the following table of values:
| x | \dfrac{1}{3} | \dfrac{2}{3} | 3 | 9 | 81 |
|---|---|---|---|---|---|
| y |
Find the x-value of the x-intercept.
How many y-intercepts does the function have?
Find the x-value for which \log_{3} x = 1.
Sketch the graphs of y = \log_{3} x and y = \log_{5} x on the same set of axes.
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 function y = \log_{\frac{1}{3}} x.
Complete the following table of values:
| x | \dfrac{1}{81} | \dfrac{1}{9} | \dfrac{1}{3} | 1 | 3 | 243 |
|---|---|---|---|---|---|---|
| y |
Is \log_{\frac{1}{3}} x an increasing or decreasing function?
Describe the behaviour of \log_{\frac{1}{3}} 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 what values of x will \log_{a} x be negative?
For what values of x will \log_{a} x 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 function y = \log_{a} x.
Determine whether the following values of a will make y = \log_{a} x an increasing function.
a = 0
a = \dfrac{1}{2}
a = 2
Determine whether the following values of a will make y = \log_{a} x a decreasing function.
a = 0
a = \dfrac{1}{2}
a = 2
Determine whether the following values of a will make y = \log_{a} x a relation, not a function.
a = e
a = 1
a = 0
When a = 1, what does the graph of the relation look like?
Describe the following transformations:
The transformation of g \left( x \right) = \log_{3} x into f \left( x \right) = \log_{3} x + k, where k > 0.
The transformation of g \left( x \right) = \log_{3} x into f \left( x \right) = \log_{3} x + k, wherek < 0.
The transformation of g \left( x \right) = \log_{10} x into f \left( x \right) = a \log_{10} x, where a > 1.
The transformation of g \left( x \right) = \log_{2} x into f \left( x \right) = a \log_{2} x, where 0 < a < 1.
The transformation of g \left( x \right) = a \log_{10} x into f \left( x \right) = - a \log_{10} x.
Consider the functions f\left(x\right) = \log_{3} x and g\left(x\right) = \log_{3} x + 3:
Complete the table of values below:
| x | \dfrac{1}{3} | 1 | 3 | 9 |
|---|---|---|---|---|
| f\left(x\right)=\log_3 x | ||||
| g\left(x\right)=\log_3 x + 3 |
Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.
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) - 2.
Complete the table of values below:
| x | -4 | -2 | -1 | -\dfrac{1}{2} |
|---|---|---|---|---|
| f\left(x\right)=\log_2 \left( - x \right) | ||||
| g\left(x\right)=\log_2 \left( - x \right) - 2 |
Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.
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.
The graph of y = \log_{6} x is transformed to create the graph of y = \log_{6} x -3. Describe a tranformation that could achieve this.
Consider the graph of y = \log_{6} x which has a vertical asymptote at x = 0. This graph is transformed to give the graph of y = \log_{6} x - 8. State the equation of the asymptote of the new graph.
Sketch the graph of the following functions:
y = \log_{2} 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 on a number plane.
y = \log_{5} x translated downwards by 2 units.
y = \log_{3} \left( - x \right) translated upwards by 3 units.
Consider the functions f \left( x \right) = 3 \log_{2} x and g \left( x \right) = \log_{2} x.
Evaluate g \left( 2 \right).
Evaluate f \left( 2 \right).
How does the graph of f \left( x \right) differ from the graph of g \left( x \right)?
Consider the graph of y = \log_{4} x which has a vertical asymptote at x = 0, state the equation of the asymptote of y = 4 \log_{4} x.
Consider the graphs of the functions f \left( x \right) = \log_{5} x and g \left( x \right):
Determine the equation of the function g \left( x \right).
Consider the graphs of the functions f \left( x \right) = \log_{2} x and g \left( x \right):
Determine the equation of the function g \left( x \right).
Consider the graph of y = \log_{9} x:
Describe the transformation of the graph of y = \log_{9} x to get the graph of y = - 2 \log_{9} x.
Hence, sketch the graph of \\y = - 2 \log_{9} x on the same set of axes as y = \log_{9} x.
Sketch the graphs of the following functions on the same set of axes:
y = \log_{6} \left( - x \right) and y = 5 \log_{6} \left( - x \right)
y = \log_{2} x and y = \dfrac{1}{3} \log_{2} x
y = \log_{2} x and y = - \dfrac{1}{3} \log_{2} x