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Magazine Chapter 5: Problem 43
Graph each function. $$f(x)=\log _{3}(x-1)$$
Short Answer
Expert verified
Graph the function with a vertical asymptote at \(x = 1\), passing through points \((2, 0.631)\), \((4, 1)\), and \((10, 2)\).
Step by step solution
01
Understand the Function
The function given is a logarithmic function of the form \(f(x) = \log_{3}(x-1)\). It represents the logarithm of \(x-1\) with base 3. This implies that the graph will have a vertical asymptote at the line \(x = 1\), since \(\log_{3}(0)\) is undefined.
02
Identify Domain and Range
The domain of the function \(f(x) = \log_{3}(x-1)\) is where the argument of the logarithm is positive. This means \(x-1 > 0\) or \(x > 1\). Hence, the domain is \((1, \infty)\) and the range is \((-\infty, \infty)\).
03
Determine Key Points
Identify key points to help with graphing. For instance, find the point where \(x-1\) equals 3 raised to a power: \(\log_{3}(2)\), \(\log_{3}(3) = 1\), and \(\log_{3}(9) = 2\). Thus, \(f(2) \approx 0.631\), \(f(4) = 1\), and \(f(10) = 2\). This gives us points \((2, 0.631), (4, 1), (10, 2)\) on the graph.
04
Plot the Vertical Asymptote
Draw a vertical dashed line at \(x = 1\), indicating a vertical asymptote where the function is undefined and the graph approaches but never touches or crosses the asymptote.
05
Plot the Key Points
Plot the identified key points on the coordinate plane: \((2, 0.631)\), \((4, 1)\), and \((10, 2)\). These will help in drawing the curve of the logarithmic function.
06
Sketch the Curve
With the vertical asymptote in place and the key points plotted, sketch the curve starting near the asymptote and passing through the key points, moving slowly upward.The graph approaches the vertical asymptote as \(x\) approaches 1 from the right and rises slowly as \(x\) increases.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Functions
Logarithmic functions, such as the one in our exercise, are inverses of exponential functions. If you understand exponents like how \(b^y = x\), then you can think of logarithms as giving you the answer "what power do we raise \(b\) to get \(x\)?"
For instance, in a logarithmic function like \(f(x) = \log_{3}(x-1)\), it essentially asks "3 raised to what power gives you \(x-1\)?"
The notation \(\log_{3}(x-1)\) means you are working with a logarithm where the base is 3 and the argument is \(x-1\).
For instance, in a logarithmic function like \(f(x) = \log_{3}(x-1)\), it essentially asks "3 raised to what power gives you \(x-1\)?"
The notation \(\log_{3}(x-1)\) means you are working with a logarithm where the base is 3 and the argument is \(x-1\).
- Key takeaway: A logarithm always asks the question about an exponent.
- The base (in this case, 3) determines the rate at which the logarithmic function increases.
Domain and Range
The domain and range of a logarithmic function determine where the function is defined and the output it can provide. Consider the function \(f(x) = \log_{3}(x-1)\) from our exercise.
The domain of this function is defined by the expression inside the logarithm, \(x-1\), being greater than zero.
This means that \(x > 1\), so the domain is \((1, \infty)\).
The domain of this function is defined by the expression inside the logarithm, \(x-1\), being greater than zero.
This means that \(x > 1\), so the domain is \((1, \infty)\).
- This domain indicates that you can only input values greater than 1 into the function.
- If \(x\) is exactly 1 or less, the logarithmic function becomes undefined since the logarithm of zero or negative numbers isn't defined in the real number system.
- No matter how far you go along the x-axis (i.e., as \(x\) increases), there will be a corresponding value for \(f(x)\).
- That's why the range is \((-\infty, \infty)\), meaning it can output any real number.
Vertical Asymptote
Vertical asymptotes are crucial features on graphs of logarithmic functions because they highlight where the function "breaks" or becomes undefined.
In the function \(f(x) = \log_{3}(x-1)\), there is a vertical asymptote at \(x = 1\).
In the function \(f(x) = \log_{3}(x-1)\), there is a vertical asymptote at \(x = 1\).
- This happens because as \(x\) approaches 1 from the right (values ">1"), the value \(x-1\) approaches 0.
- You cannot take the logarithm of zero, so the function approaches infinity as it gets closer to \(x = 1\).
- This results in the "wall" or vertical line at \(x = 1\) where the function cannot pass.
- It is typically represented as a dashed line on the graph.
- It serves as a barrier that the graph approaches but never crosses or truly touches.
Key Points in Graphing
Identifying key points on a graph helps to properly sketch the curve of a function, especially for translating abstract equations into visual shapes.
For the function \(f(x) = \log_{3}(x-1)\), important points are those where the function changes significantly or hits recognizable numbers.
For the function \(f(x) = \log_{3}(x-1)\), important points are those where the function changes significantly or hits recognizable numbers.
- Calculate points by substituting values for \(x\) which solve simple equations like \(3^0, 3^1, 3^2\), evaluating the logarithmic function for these x-values gives recognizable y-values.
- Example: The points \((2, 0.631), (4, 1), (10, 2)\) were derived by inputting \(x\)-values and solving \(\log_3(\text{value})\).
- These key points offer clues on how to draw the curve: starting near the vertical asymptote and then gradually rising as \(x\) increases.
- The graph becomes easier to understand and predict.
- Use these plotted points to guide the drawing of the curve, ensuring it reflects the smooth and non-linear nature of logarithmic growth.
