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Chapter 10: Problem 92

Graph each logarithmic function. $$f(x)=\log _{1 / 5} x$$

Short Answer

Expert verified
Plot points (5, -1), (1, 0), and (\( \frac{1}{5} \), 1) and draw a decreasing curve, approaching \(x = 0\) asymptotically.

Step by step solution

01

Understand the Function

The given function is \(f(x) = \log_{1/5} x\). This notation means that we are dealing with a logarithm with base \(\frac{1}{5}\). Understanding that this base is less than 1 is crucial because it affects the shape and direction of the graph.
02

Determine Key Points

To graph \(f(x) = \log_{1/5} x\), select key points by converting the logarithmic form \(y = \log_{1/5} x\) into the exponential form: \(x = \left(\frac{1}{5}\right)^y\). Pick values for \(y\) and solve for \(x\). For example, when \(y = -1\), \(x = 5\); when \(y = 0\), \(x = 1\); when \(y = 1\), \(x = \frac{1}{5}\).
03

Create a Table of Values

Construct a table with the chosen key points:\[\begin{array}{c|c}y & x \ \hline-1 & 5 \ 0 & 1 \ 1 & \frac{1}{5} \ \end{array}\]
04

Plot the Points

On a coordinate plane, plot the points from your table: (5, -1), (1, 0), and (\( \frac{1}{5} \), 1).
05

Draw the Curve

Draw a smooth curve through the points, remembering that the logarithmic function concerning the base \(\frac{1}{5}\) will decrease as \(x\) increases. The graph will approach the vertical asymptote at \(x = 0\) but will never touch or cross it.

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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 mathematical function defined as the inverse of an exponential function. In simpler terms, if you have an exponential function such as \( b^y = x \), the logarithmic form would be \( y = \log_b (x) \). This tells us that \( y \) is the exponent to which the base \( b \) must be raised to get \( x \).
For example, in \( f(x) = \log_{1/5} x \):
  • The base is \( \frac{1}{5} \)
  • \( f(x) \) is the exponent.
Understanding this helps us to graph the function correctly as each input \( x \) relates to an exponent necessary to get the base to equal \( x \). Remember, logarithmic functions are defined only for positive inputs (\( x > 0 \)).
base of a logarithm
The base of a logarithm is the number that is repeatedly multiplied by itself to produce a given value. In the function \( f(x) = \log_{1/5} x \), the base is \( \frac{1}{5} \). This means:
  • To find the logarithm of a number, we determine the power (or exponent) we must raise \( \frac{1}{5} \) to, in order to get that number.
Since \( \frac{1}{5} \) is less than 1, the logarithm will have properties different from those functions with bases greater than 1. Specifically, the function \( \log_{1/5} x \) will be decreasing, meaning it goes down as \( x \) increases. The fundamental idea is to understand how changing the base affects the graph's direction and steepness.
exponential form
Converting a logarithmic equation to an exponential form simplifies plotting key points on the graph.
For the function \( y = \log_{1/5} x \), we convert it to its exponential form: \( x = \left( \frac{1}{5} \right)^y \).
  • This helps us identify key points (\( x, y \)) for graphing.
For instance, if \( y = -1 \), then \( x = 5 \) since \( \left( \frac{1}{5} \right)^{-1} = 5 \). Similarly, if \( y = 0 \), \( x = 1 \), and if \( y = 1 \), \( x = \frac{1}{5} \).
By converting back and forth between logarithmic and exponential forms, we can gather enough data points to plot an accurate graph.
vertical asymptote
A vertical asymptote is a line that a graph approaches but never touches or crosses. For logarithmic functions like \( f(x) = \log_{1/5} x \), the vertical asymptote is found where the function does not exist.
In this case:
  • The function does not exist for \( x \leq 0 \).
Therefore, the vertical asymptote is at \( x = 0 \). When graphing, notice how the curve gets closer and closer to this line without ever touching it. This behavior exemplifies the idea that logarithms of zero or negative numbers are undefined in real numbers, creating a boundary in the graph.

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Most popular questions from this chapter

Find the amount of money in an account after 8 yr if \(\$ 4500\) is deposited at \(6 \%\) annual interest compounded as follows. (a) Annually (b) Semiannually (c) Quarterly (d) Daily (Use \(n=365 .)\) (e) Continuously

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} 162 $$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} \frac{1}{9} $$

Revenues of software publishers in the United States for the years \(2004-2016\) can be modeled by the function $$ S(x)=91.412 e^{0.05195 x} $$ where \(x=4\) represents \(2004, x=5\) represents \(2005,\) and so on, and \(S(x)\) is in billions of dollars. Approximate, to the nearest unit, revenue for \(2016 .\) (Data from U.S. Census Bureau.)

Solve each equation. Approximate solutions to three decimal places. $$ 2^{x+3}=3^{x-4} $$
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