Problem 9
Question
Draw the graphs of $$ f(x)=\log _{2}(\sin x) $$
Step-by-Step Solution
Verified Answer
The graph of \( f(x) = \log_2(\sin x) \) is defined only where the value of the sine function is greater than 0, i.e., in intervals \((0, \pi)\), \((2\pi, 3\pi)\), and so on. The graph copies the general shape of the sine function, but only exists in certain intervals. It hits a maximum at points where the sine function equals 1 and becomes undefined at points where the sine function equals 0.
1Step 1: Determine Function Features
The function represents a combination of a logarithmic function and a trigonometric function. The sine function oscillates between -1 and 1, with a period of \(2\pi\). The logarithmic function, \(log_2x\), is only defined for \(x > 0\). Thus, the original function, \(f(x) = \log_2(\sin x)\), exists only where \(\sin x > 0\).
2Step 2: Interval for Existence of Function
From the above, we see that \(\sin x > 0\) in the intervals \((0, \pi)\) and \((2\pi, 3\pi)\) and so on. These are the intervals where the function \(f(x) = \log_2(\sin x)\) will exist and we should only consider these intervals when plotting the graph.
3Step 3: Function Values
For the intervals where the function exists, the next step is to find the function values. Knowing that \(log_2(1) = 0\) and \(log_2(0.5) = -1\), this gives us two points for each period of the sine function.
4Step 4: Function Graph
Lastly, having established the intervals of existence of the function and the function values, we are now in a position to graph the function. Begin by graphing the sine function, then mark off the intervals where the function is not defined, i.e., where \(\sin x \leq 0\). Draw the curve through the above points, reproducing the pattern for each period. Also mark off when the points hit a maximum.
Key Concepts
Logarithmic Function PropertiesTrigonometric Function OscillationSine Function Period
Logarithmic Function Properties
Understanding the properties of logarithmic functions plays a crucial role in graphing functions like \( f(x)=\log_{2}(\sin x) \). Logarithmic functions are the inverse of exponential functions. One of the key features of a logarithmic function is that it is only defined for positive real numbers. This is why we talk about the logarithm of a positive number, never of zero or negative numbers.
The base of the logarithm also affects the shape of the graph. For instance, the function \( \log_{2}(x) \) has a base of two and will increase more slowly compared to a function with a larger base. It's also important to remember that the logarithmic function passes through the point \( (1,0) \), since \( \log_{2}(1) \) equals zero regardless of the base.
When graphing logarithmic functions, it can be helpful to recognize that as the x-values approach zero from the positive side, the y-values decrease without bound—this illustration is referred to as a vertical asymptote.
The base of the logarithm also affects the shape of the graph. For instance, the function \( \log_{2}(x) \) has a base of two and will increase more slowly compared to a function with a larger base. It's also important to remember that the logarithmic function passes through the point \( (1,0) \), since \( \log_{2}(1) \) equals zero regardless of the base.
When graphing logarithmic functions, it can be helpful to recognize that as the x-values approach zero from the positive side, the y-values decrease without bound—this illustration is referred to as a vertical asymptote.
Trigonometric Function Oscillation
The oscillation of trigonometric functions refers to their wave-like behavior. Functions like the sine and cosine have specific patterns that they repeatedly follow. For a sine function, this oscillation occurs between the values of -1 and 1. This property is especially important for functions that are a composite of a trigonometric function with another function, like our example \( f(x) = \log_2(\sin x) \).
Moreover, a characteristic feature of these oscillating functions is their amplitude, which determines the height of the peaks and the depth of the valleys. For the standard sine function, this is consistently 1. Additionally, the trigonometric functions are periodic, which means they repeat their values in regular intervals, a property that is known as periodicity and is measured by the period of the function.
Moreover, a characteristic feature of these oscillating functions is their amplitude, which determines the height of the peaks and the depth of the valleys. For the standard sine function, this is consistently 1. Additionally, the trigonometric functions are periodic, which means they repeat their values in regular intervals, a property that is known as periodicity and is measured by the period of the function.
Sine Function Period
The period of the sine function is the length of one complete cycle of its oscillation. For the standard sine function, \(\sin(x)\), this period is \(2\pi\) radians. This periodic attribute is essential when graphing trigonometric functions as it helps determine the repeating intervals.
In our exercise with \( f(x)=\log_{2}(\sin x) \), understanding the period of the sine function informs us that the pattern of the graph will repeat every \(2\pi\) units along the x-axis. Because the logarithmic function only takes positive inputs, we only graph the portion of the sine wave where the sine value is positive—that is, from \(0\) to \(\pi\), from \(2\pi\) to \(3\pi\), and so on. Each of these intervals constitutes half a period of the sine function, which demonstrates how the sine function's period influences the graph of the combined function.
In our exercise with \( f(x)=\log_{2}(\sin x) \), understanding the period of the sine function informs us that the pattern of the graph will repeat every \(2\pi\) units along the x-axis. Because the logarithmic function only takes positive inputs, we only graph the portion of the sine wave where the sine value is positive—that is, from \(0\) to \(\pi\), from \(2\pi\) to \(3\pi\), and so on. Each of these intervals constitutes half a period of the sine function, which demonstrates how the sine function's period influences the graph of the combined function.
Other exercises in this chapter
Problem 8
Draw the graphs of $$ f(x)=x \cos x $$
View solution Problem 8
Draw the graphs of $$ f(x)=\sin \left(x+\frac{\pi}{6}\right) $$
View solution Problem 9
Draw the graphs of $$ f(x)=x+\cos x $$
View solution Problem 9
Draw the graphs of $$ f(x)=\sin (x-1) $$
View solution