2.11 Logarithmic functions
What is the domain and range of a logarithmic function in general form?
Determine whether the statements are true or false:
The logarithmic function is increasing if the base is greater than 1.
Logarithmic functions have a vertical asymptote.
The graphs of logarithmic functions always have points of inflection.
Logarithmic functions have extrema.
What is the end behavior of a logarithmic function a\log x form in your answer.
What happens to the input values of the additive transformation function g (x) = f (x + k)?
If the input values of the additive transformation function g (x) = f (x + k) are proportional over equal-length output value intervals, then what type of function is f?
Determine the value of \lim_{x \to 0^{+}} a\log_b x given the condition:
b \gt 1, \, a \gt 0
b \gt 1, \, a \lt 0
0 \lt b \lt 1, \, a \gt 0
0 \lt b \lt 1, \, a \lt 0
For each logarithmic functions:
Find the domain and the range.
Identify the behavior of the function as x approaches 0.
Is the function's graph likely to be concave up, concave down, increasing or decreasing?
Identify any points of inflection for the function.
f \left(x\right)=- \log \left(-3x\right)
f \left(x\right)= - \log \left(x^2\right)
The function f\left(x\right)=\log_{7}\left(x\right) is transformed into g\left(x\right)=f\left(x-6\right).
Write the equation for g\left(x\right).
What type of transformation has occurred from f(x) to g(x)?
What is the effect of this transformation on the input values of f(x)?
Determine if the input values of g\left(x\right) are proportional over equal-length output value intervals.
Consider the function p(x) = x^2 and its transformation q(x) = p(x + 3).
Is the relationship between the input values of the two functions proportional over equal-length output value intervals? Justify your answer.
What does this tell us about the nature of the function p(x)?
The table provides some values of a logarithmic function.
Identify the base of the logarithm used in the function.
Is the function's graph likely to be concave up or concave down?
Is the function increasing or decreasing?
Sketch the graph of the function.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 0 | 0.3 | 0.48 | 0.6 | 0.7 |
Here is the graph of the function y = \log_{2}(x).
Determine the behavior of the function as x approaches positive infinity and as x approaches zero from the right.
Does the function have any extrema or points of inflection? Explain your reasoning.
For the function f \left( x \right) = 3 \cdot \log\left(x - 2\right), determine the behavior of the function as x approaches 0. Does the function approach positive or negative infinity, or is it undefined? Explain your answer.
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) = - \log_{4} \left(x + 4\right)
f \left(x\right) = \log_{2} \left(x - 1\right) - 4
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.
Find the limit as x approaches 0 from the right for the function:
f(x) = \log_{e}(x)
g(x) = \log_{10}(x)
h(x) =\log_{2}(x)
k(x) = -log_{5}(x)
p(x) = 2\log_{10}(x)
q(x) = 3\log_{2}(x)
r(x) =-4\log_{3}(x)
s(x) = 5\log_{4}(x)
A function is defined by the equation f \left( x \right)= \log_{4} (x^2 - 3x + 2).
What is the domain of the function?
What is the range of the function?
What are the asymptotes of the function?
How would the domain and range change if the base of the logarithm was changed to 2?
The function h(x) = \log_{4}(x^2 - 1) is a transformation of the parent function y = \log_{b}(x).
Identify the transformation that has been applied to the parent function to obtain the given function.
Sketch the graph of the function.
Describe its behavior as x approaches negative infinity and as x approaches positive infinity.
Does the function have any extrema or points of inflection? Explain your reasoning.
Is the graph concave up or concave down? Justify your answer.
Consider the function f(x) = 2x + 3 and its transformation g(x) = f(x + k), where k is a constant.
Explain how the value of k would affect the transformation of the function f(x).
If the input values of the function g(x) are proportional over equal-length output value intervals, what can you conclude about the nature of the function f(x)?
If the function f(x) is not a logarithmic function, can the function g(x) be a logarithmic function? Justify your answer.
The function g \left( x \right) = \log\left(x\right) + 2 models the pH level of a solution based on the concentration of hydrogen ions, x.
As the concentration of hydrogen ions approaches 0, what happens to the pH level of the solution?
Is the solution becoming more acidic, more basic, or is it neutral?
Based on the properties and behavior of logarithmic functions, explain why the pH level changes in the way you described as the concentration of hydrogen ions approaches 0. Consider the domain and range of the function g \left( x \right) = \log\left(x\right) + 2 in your explanation.
A sound engineer is measuring the loudness of a sound over time. The loudness L in decibels after t seconds is modeled by the logarithmic function L \left( t \right)=10\log_{10}(t+1).
Identify the domain and range of this function. Describe what these represent in the context of the situation.
Describe the end behavior of the function. What does this represent in the context of the situation?
The engineer makes an adjustment to the microphone and now the loudness is modeled by the function M \left( t \right)=10\log_{10}(t+2). How does this affect the loudness measurements?
Determine if the input values of the transformed function are proportional over equal-length output value intervals. What does this tell you about the original function?