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Magazine Chapter 4: Problem 10
Sketch a graph of each pair of function $$ f(x)=\log _{2}(x), g(x)=\log _{4}(x) $$
Short Answer
Expert verified
Both functions are increasing for \( x > 0 \), with \( f(x) \) steeper than \( g(x) \).
Step by step solution
01
Understand Logarithmic Functions
The functions given, \( f(x) = \log_{2}(x) \) and \( g(x) = \log_{4}(x) \), are logarithmic functions. The base of the logarithm indicates the number for which successive powers will equal \( x \). For \( f(x) \), the base is 2, while for \( g(x) \), the base is 4.
02
Determine the Domain
For both functions, \( f(x) = \log_{2}(x) \) and \( g(x) = \log_{4}(x) \), the domain is \( x > 0 \). This means that both functions are only defined for positive values of \( x \).
03
Identify Key Points
For \( f(x) = \log_{2}(x) \), determine key points such as \( f(1) = 0 \) and \( f(2) = 1 \) since \( \log_{2}(2) = 1 \). For \( g(x) = \log_{4}(x) \), similarly find \( g(1) = 0 \) and \( g(4) = 1 \) because \( \log_{4}(4) = 1 \).
04
Understand the Behavior
Both logarithmic functions are increasing on their domain. As \( x \rightarrow 0^{+} \), both functions tend to negative infinity; as \( x \rightarrow \infty \), the functions increase steadily.
05
Compare the Growth Rates
\( f(x) = \log_{2}(x) \) grows faster than \( g(x) = \log_{4}(x) \). This is because the base of \( f(x) \) is smaller, making \( f(x) \)'s increase sharper over equivalent intervals of \( x \).
06
Sketch the Graphs
Plot the points identified and sketch \( f(x) \) and \( g(x) \) on the same coordinate plane. Start both at \( (1, 0) \), note that \( f(2) = 1 \) and \( g(4) = 1 \) to guide the shape. Remember, \( f(x) \) climbs more steeply than \( g(x) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain of Functions
When dealing with logarithmic functions like \( f(x) = \log_{2}(x) \) and \( g(x) = \log_{4}(x) \), understanding the domain is crucial. The domain of a function is the set of all possible input values (\( x \) values) for which the function is defined. In the case of logarithms, we know that we can only take the logarithm of positive numbers. This means the domain for both functions is \( x > 0 \).
No negative numbers or zero can be part of the domain, because logarithms of non-positive values are undefined.
No negative numbers or zero can be part of the domain, because logarithms of non-positive values are undefined.
- For any logarithmic function \( \log_{a}(x) \), \( x \) must always be greater than zero.
- Remember to check that the base of the logarithm, in this case, 2 and 4, remains positive and not equal to 1, which are beautiful properties that must be upheld.
Graphing Techniques
Graphing logarithmic functions requires a keen understanding of their behavior and certain techniques to accurately represent them visually. Begin by plotting key points to serve as anchors for the graph. For \( f(x) = \log_{2}(x) \), plot points like \( (1, 0) \) and \( (2, 1) \). For \( g(x) = \log_{4}(x) \), use \( (1, 0) \) and \( (4, 1) \).
These points help ensure that the shape of the graph is correct, particularly since logarithms have a distinct curve that starts steep and flattens.
These points help ensure that the shape of the graph is correct, particularly since logarithms have a distinct curve that starts steep and flattens.
- Both functions will pass through the point \( (1, 0) \) - a shared characteristic of any \( \log_{a}(x) \) function.
- As \( x \) increases, the curves climb but begin to flatten, so the graph should show a slower ascent as \( x \) moves away from the initial key points.
- Remember the vertical asymptote at \( x = 0 \); the graph never touches the y-axis but gets infinitesimally close as \( x \) approaches zero.
Exponential Growth Rates
Understanding the growth rate of logarithmic functions is integral to comprehending their behavior. These functions exhibit a slow, steady increase, but how rapidly they grow is dictated by their base. For \( f(x) = \log_{2}(x) \), the base is 2, and for \( g(x) = \log_{4}(x) \), it's 4.
Logarithmic growth contrasts with exponential growth; while both increase, logarithmic growth is much slower and flattens as x becomes large.
Logarithmic growth contrasts with exponential growth; while both increase, logarithmic growth is much slower and flattens as x becomes large.
- Smaller bases lead to steeper initial increases, which is why \( \log_{2}(x) \) seems to grow faster than \( \log_{4}(x) \).
- In the plot, the steepness reflects how quickly logs of numbers increase, highlighting the visual difference between base 2 and base 4.
- This concept is crucial in fields like computer science, where understanding logarithmic versus exponential time complexities affects algorithm efficiency.
Key Points Identification
Identifying key points on the graph of a logarithmic function is a strategic step in constructing or analyzing the graph. These points give precise locations where the graph changes or has specific values. For \( f(x) = \log_{2}(x) \), notable points include \( (1, 0) \) and \( (2, 1) \). These points correspond to the fact that \( 2^0 = 1 \) and \( 2^1 = 2 \).
For \( g(x) = \log_{4}(x) \), similar logic gives you points like \( (1, 0) \) and \( (4, 1) \), based on the relationships \( 4^0 = 1 \) and \( 4^1 = 4 \).
For \( g(x) = \log_{4}(x) \), similar logic gives you points like \( (1, 0) \) and \( (4, 1) \), based on the relationships \( 4^0 = 1 \) and \( 4^1 = 4 \).
- Key points help guide the smooth curve of the graph, ensuring it aligns with known values.
- Beyond these initial points, one can add more based on other powers for deeper analysis of the graph shape.
- Knowing these points simplifies graph sketching and ensures the scaling and axis intersections are correct.
