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Chapter 10: Problem 91

Graph each logarithmic function. $$f(x)=\log _{1 / 3} x$$

Short Answer

Expert verified
Plot points (1, 0), (3, -1), (1/3, 1), and draw the vertical asymptote at \(x = 0\).

Step by step solution

01

Understand the Function Form

The given function is a logarithmic function of the form \(f(x) = \log_{a}(x)\) where the base \(a\) is \(\frac{1}{3}\).
02

Identify Key Points

Find key points for the logarithmic function. For \(f(x) = \log_{1/3}(x)\), select key points: * When \(x = 1\), \(f(x) = \log_{1/3}(1) = 0\) * When \(x = 3\), \(f(x) = \log_{1/3}(3) = -1\) * When \(x = 1/3\), \(f(x) = \log_{1/3}(1/3) = 1\).
03

Determine Asymptote

Logarithmic functions have vertical asymptotes. Since \(f(x) = \log_{1/3}(x)\), the vertical asymptote is at \(x = 0\).
04

Plot the Key Points and Asymptote

Mark the points (1, 0), (3, -1), and (1/3, 1) on the graph. Also, draw a vertical line at \(x = 0\) to represent the asymptote.
05

Draw the Curve

Draw the curve passing through the points and approaching the vertical asymptote at \(x = 0\). Make sure the curve demonstrates the nature of being a decreasing function for \(x > 0\).

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Key Concepts

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

logarithmic functions
A logarithmic function is an inverse operation to exponentiation. It gives you the exponent as a result. For example, the function \(f(x) = \log_{a}(x)\) asks, 'To what power must we raise a to get x?'. The base \(a\) must be positive and cannot be 1. In this exercise, the base is \(\frac{1}{3}\), which is a fraction. When graphing, it's vital to understand how different bases affect the shape of the logarithmic curve.
  • If \(0 < a < 1\), the graph decreases.
  • If \(a > 1\), the graph increases.
For our function, \(\log_{1/3}(x)\), the graph is decreasing since \(0 < 1/3 < 1\). This behavior will become clearer when we plot the key points.
decreasing functions
If a function is decreasing, it means that as the \(x\) values increase, the \(y\) values decrease. For our function \(f(x) = \log_{1/3}(x)\), the base, \(1/3\), is less than 1. Hence, as \(x\) gets larger, the \(y\) values will become smaller. This is evident from the points we found earlier.
  • At \(x = 3\), \(f(x) = -1\)
  • At \(x = 1\), \(f(x) = 0\)
  • At \(x = 1/3\), \(f(x) = 1\)
The function decreases across these points, showing the nature of the logarithm with a fractional base. When drawing the graph, ensure the curve is tight to the vertical asymptote and smoothly decreases through the key points.

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Most popular questions from this chapter

Solve each problem. Sales (in thousands of units) of a new product are approximated by the logarithmic function $$S(t)=100+30 \log _{3}(2 t+1)$$ where \(t\) is the number of years after the product is introduced. (a) What were the sales, to the nearest unit, after 1 yr? (b) What were the sales, to the nearest unit, after 13 yr? (c) Graph \(y=S(t)\)

Solve each equation. Approximate solutions to three decimal places. $$ 4^{x-2}=5^{3 x+2} $$

Solve each equation. Approximate solutions to three decimal places. $$ 9^{-x+2}=13 $$

Based on selected figures obtained during the years \(1970-2015,\) the total number of bachelor's degrees earned in the United States can be modeled by the function $$ D(x)=792,377 e^{0.01798 x} $$ where \(x=0\) corresponds to \(1970, x=5\) corresponds to \(1975,\) and so on. Approximate, to the nearest unit, the number of bachelor's degrees earned in 2015. (Data from U.S. National Center for Education Statistics.)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{\ln 2 x}=e^{\ln (x+1)} $$
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