/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 185 In the following exercises, grap... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 185

In the following exercises, graph each logarithmic function. \(y=\log _{\frac{1}{5}} x\)

Short Answer

Expert verified
Plot points \((1, 0)\), \((\frac{1}{5}, 1)\), and \((5, -1)\). Draw the curve through these points.

Step by step solution

01

Understand the function

The function given is a logarithmic function of the form \(y = \log _{ \frac{1}{5}} x\). Here, the base of the logarithm is \( \frac{1}{5}\).
02

Identify key points

Key points for graphing logarithmic functions are generally found by using simple values for \(x\) that make the logarithm easier to evaluate. Start with values like \(x = 1, x = \frac{1}{5}, x = 5\).
03

Calculate the key points

For \(x = 1\), \(y = \log _{\frac{1}{5}} 1 = 0\). For \(x = \frac{1}{5}\), \(y = \log _{\frac{1}{5}} \frac{1}{5} = 1\). For \(x = 5\), \(y = \log _{\frac{1}{5}} 5 = -1\).
04

Plot the points

Plot the points \((1, 0)\), \((\frac{1}{5}, 1)\), and \((5, -1)\) on the graph.
05

Draw the curve

Draw a smooth curve through these points. Remember that the graph of a logarithmic function approaches the y-axis (vertical asymptote) but never touches it.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

logarithmic function
A logarithmic function is an inverse of an exponential function. If you understand exponents, then you are already halfway to understanding logarithms. In a logarithmic function of the form \( y = \log_{b} x \), the base \( b \) is always a positive number, not equal to one, and \( x \) is a positive number. For example, \( y = \log_{\frac{1}{5}} x \) means we're looking for a number \( y \) such that \( (\frac{1}{5})^y = x \). For every positive \( x \), there's a corresponding \( y \) value that satisfies this equation.
graphing techniques
When graphing logarithmic functions, it's essential to identify some key points that belong to the function. Start by picking points for \( x \) that make the logarithm calculations simple. Typical key points include 1, the base \( b \), and the reciprocal of the base \( \frac{1}{b} \). For example, with the function \( y = \log_{\frac{1}{5}} x \), you would choose \( x = 1 \), \( x = 5 \), and \( x = \frac{1}{5} \). Then, calculate the corresponding \( y \) values:
  • For \( x = 1 \), \( y = \log_{\frac{1}{5}} 1 = 0 \).
  • For \( x = \frac{1}{5} \), \( y = \log_{\frac{1}{5}} \frac{1}{5} = 1 \).
  • For \( x = 5 \), \( y = \log_{\frac{1}{5}} 5 = -1 \).
Plot these points on the graph and draw a smooth and continuous curve through them.
logarithm properties
Understanding the properties of logarithms can make graphing and manipulating these functions easier. Some critical properties are:
  • \( \log_b 1 = 0 \) for any base \( b \), because any number raised to the power of 0 is 1.
  • \( \log_b b = 1 \), reflecting that raising the base \( b \) to the power of 1 returns \( b \).
  • Logarithms convert multiplication into addition: \( \log_b (xy) = \log_b x + \log_b y \).
  • Logarithms convert division into subtraction: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
  • Exponents inside the log can be moved to the front: \( \log_b (x^k) = k \log_b x \).
Each of these properties simplifies working with logarithmic functions.
vertical asymptote
One crucial feature of the graph of a logarithmic function is the vertical asymptote. This is a line that the graph approaches but never touches or crosses. For \( y = \log_b x \), the vertical asymptote is at \( x = 0 \). This happens because logarithmic functions are undefined for non-positive values of \( x \). For example, in the function \( y = \log_{\frac{1}{5}} x \), as \( x \) gets closer to 0 from the right, \( y \) decreases without bound (\( y \) approaches negative infinity). Knowing the location of the vertical asymptote helps you sketch the graph more accurately by ensuring the curve approaches but does not cross the \( y \)-axis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log x^{-3}\)

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm. \(\log _{12} 87\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(4 \log x-2 \log y-3 \log z\)

In the following exercises, solve for \(x\). \(\log _{5}(4 x-2)=\log _{5} 10\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(3 \ln x+4 \ln y-2 \ln z\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.