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Chapter 12: Problem 42

Graph by hand. $$ f(x)=\log _{4} x $$

Short Answer

Expert verified
Plot points (1,0), (4,1), (16,2) and draw a curve that approaches but never touches the y-axis; label the graph.

Step by step solution

01

Understand the Function

The function given is a logarithmic function: \( f(x) = \log_{4}x \). This means that for any value of \( x \), the function returns the exponent to which 4 must be raised to get \( x \).
02

Identify Key Points

To graph \( f(x) = \log_{4}x \), identify some key points: 1. \( f(1) = \log_{4}1 = 0 \) because \( 4^0 = 1 \) 2. \( f(4) = \log_{4}4 = 1 \) because \( 4^1 = 4 \) 3. \( f(16) = \log_{4}16 = 2 \) because \( 4^2 = 16 \) These points help sketch the overall shape of the graph.
03

Set Up the Coordinate Plane

Draw a coordinate plane with the x-axis representing the values of \( x \) and the y-axis representing the values of \( f(x) \). Label the axes appropriately and mark a reasonable range for the x and y values based on the key points identified.
04

Plot Key Points

Plot the key points identified in Step 2 onto the coordinate plane: - (1, 0) - (4, 1) - (16, 2)
05

Draw the Curve

Draw a smooth curve that passes through the plotted points. Make sure the curve approaches the y-axis but never touches it, showing that \( \log_{4}(x) \) becomes very large and negative as \( x \) approaches zero.
06

Add Asymptote

Since \( \log_{4}x \) is undefined for \( x \leq 0 \), show that there is a vertical asymptote along the y-axis (x = 0). The graph will approach this line but never touch or cross it.
07

Label the Graph

Finally, label the graph of the function as \( f(x) = \log_{4}x \). Ensure the points and asymptote are clearly marked to complete the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Function
A logarithmic function is a type of function where the exponent is the variable. For example, in the function \( f(x) = \log_{4}x \), \( x \) is the argument of the logarithm with base 4. Logarithmic functions commonly represent exponential growth and decay in real-world scenarios.
Understanding the basic properties of the logarithm can help you comprehend these functions:
  • Logarithms undo exponentiation, meaning they find the exponent that a base must be raised to produce a certain number.
  • The logarithmic function \( f(x) = \log_{b}x \) is only defined for positive \( x \) values, where \( b \) is the base and must be greater than 0 and not equal to 1.
  • For any base \( b \), \( \log_b(1) = 0 \) because \( b^0 = 1 \).
These principles form the foundation for graphing and understanding logarithmic functions.
Coordinate Plane
To graph a logarithmic function like \( f(x) = \log_{4}x \), you will use a coordinate plane, which has two perpendicular axes:
  • The x-axis (horizontal) where you plot the values of \( x \).
  • The y-axis (vertical) where you plot the corresponding function values \( f(x) \).
When setting up your coordinate plane:
  • Start by drawing the x-axis and y-axis and marking them with regular intervals.
  • Label each axis to clarify which variable it represents.
  • Choose a reasonable range for both axes based on the function and its key points. For example, if one of your key points is \( (1, 0) \), make sure your x-axis includes 1 and your y-axis includes 0.
  • Mark important points and intervals that will help to accurately plot the graph of the function.
Setting up the coordinate plane properly helps visualize graphs and interpret relationships between variables.
Vertical Asymptote
A vertical asymptote is a line that a graph approaches but never actually touches or crosses.
For logarithmic functions, vertical asymptotes occur where the function is undefined. In the case of \( f(x) = \log_{4}x \), the function is undefined for \( x \leq 0 \).
This means we have a vertical asymptote at \( x = 0 \). Graphically, you will:
  • Draw a dashed or dotted vertical line at \( x = 0 \).
  • Make sure the curve you draw does not touch or cross this line.
The graph approaches this line as \( x \) approaches 0 from the right but never actually intersects it. Remember, the function value becomes very large and negative as \( x \) gets close to 0.
Key Points
Key points are specific coordinates that help in sketching the graph of a function. For \( f(x) = \log_{4}x \), you can identify key points by plugging in values for \( x \) to find corresponding \( y \) values:
  • \( (1, 0) \)
  • \( (4, 1) \)
  • \( (16, 2) \)
Each of these points corresponds to an x-value and their exponential equivalents:
  • For \( x=1 \), \( 4^0 = 1 \), giving the point \( (1, 0) \).
  • For \( x=4 \), \( 4^1 = 4 \), giving the point \( (4, 1) \).
  • For \( x=16 \), \( 4^2 = 16 \), giving the point \( (16, 2) \).
These points will help guide the shape of the curve on the coordinate plane. Plot them accurately, then draw a smooth curve through them that follows the characteristics of a logarithmic function.

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Most popular questions from this chapter

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} \frac{1}{5}$$

Classify each of the following as true or false. Assume a, \(x, P,\) and \(Q>0, a \neq 1\). $$\log _{a}\left(Q+Q^{2}\right)=\log _{a} Q+\log _{a}(Q+1)$$

The supply and demand for the sale of stereos by Sound Ideas are given by $$ S(x)=e^{x} \quad \text { and } \quad D(x)=162,755 e^{-x} $$ where \(S(x)\) is the price at which the company is willing to supply \(x\) stereos and \(D(x)\) is the demand price for a quantity of \(x\) stereos. Find the equilibrium point.

Use the properties of logarithms to find each of the following. $$\log _{5}(125 \cdot 625)$$

Show that for exponential growth at rate \(k,\) the doubling time \(T\) is given by \(T=\frac{\ln 2}{k}\)
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