/*! 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 38 Graph each logarithmic function.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 38

Graph each logarithmic function. $$ y=\log _{5} x+1 $$

Short Answer

Expert verified
The graph of the function \(y=\log _{5} x+1\) is the same as the basic logarithm graph \(y=\log _{5} x\), but shifted 1 unit upwards.

Step by step solution

01

Identify the Basic Function

The logarithm function given is \(y=\log _{5} x+1\). Now, one must identify the basic form of this function, which is \(y=\log _{5} x\). This is the function we first graph before considering the effects of the +1 term.
02

Graph the Basic Logarithm Function

To keep it simple, start with the points at \(x = 1\) and \(x = 5\) for the basic log graph. For \(x = 1\), the value of \(y = \log_{5}1 = 0\), while for \(x = 5\), \(y = \log_{5}5 = 1\). Draw a curve through these points.
03

Applying the Vertical Shift

The given function has a +1 added to the base logarithm function, which represents a shift 1 unit upward. Every point on the graph of \(y=\log _{5} x\) should therefore be moved 1 unit upwards to form the graph of \(y=\log _{5} x+1\). For instance, the point (1, 0) on the initial graph becomes (1, 1), and the point (5, 1) becomes (5, 2). Draw the new graph with shifted points.

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.

Vertical Shift
To understand the concept of a vertical shift in logarithmic functions, imagine taking a graph and moving it upward or downward without altering the shape. The function \( y = \log_{5} x + 1 \) demonstrates a vertical shift. Here, the "+1" indicates that every point on the graph of the basic logarithm function \( y = \log_{5} x \) moves up by 1 unit.

For example, consider the point \((1, 0)\) on \( y = \log_{5} x \). By shifting vertically, it becomes \((1, 1)\) on the graph of \( y = \log_{5} x + 1 \). Similarly, \((5, 1)\) transforms to \((5, 2)\).

Key Points of Vertical Shift:
  • Addition results in upward shifts.
  • Subtraction results in downward shifts.
  • The graph鈥檚 overall shape remains unchanged; only its position is affected.
Graphing Transformations
Graphing transformations help in visualizing how functions change in form. They encompass shifts, stretches, compressions, and reflections.

For \( y = \log_{5} x + 1 \), the primary transformation involved is the vertical shift. Even though no stretching or compression occurs here, understanding these transformations is essential for mastery.

Types of Graphing Transformations:
  • Vertical Shifts: Move the graph higher or lower.
  • Horizontal Shifts: Shift the graph left or right along the x-axis.
  • Reflections: Flip the graph over a specific axis.
  • Vertical Stretches/Compressions: Alter the graph鈥檚 steepness.
  • Horizontal Stretches/Compressions: Modify the graph鈥檚 width.

Mastering these transformations empowers you to readily adapt any function to its new form on the graph.
Base of Logarithm
Understanding the base of a logarithm is crucial, as it influences the shape of the graph. The base indicates how the output values respond to changes in the input value.

In the function \( y = \log_{5} x + 1 \), the base is 5. This base affects the graph鈥檚 steepness and rate of increase. Logarithms with larger bases grow more gradually.

Key Highlights of Logarithm Base:
  • A base greater than one leads to a positive curve.
  • The graph crosses the x-axis at \( x = 1 \), regardless of the base.
  • Different bases result in distinct graph slopes and placements.

The base forms the foundation of understanding logarithms and their associated transformations.

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

Which expression is equal to \(\log _{5} x+4 \cdot \log _{5} y-2 \cdot \log _{5} z ?\) \(\begin{array}{llll}{\text { A. } \log _{5}(-8 x y z)} & {\text { B. }-\log _{5} \frac{4 x y}{2 z}} & {\text { C. } \log _{5} \frac{(x y)^{4}}{z^{2}}} & {\text { D. } \log _{5} \frac{x y^{4}}{z^{2}}}\end{array}\)

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 9^{2 x}=42 $$

Mental Math Solve each equation. $$ 2^{x}=\frac{1}{2} $$

Which expression is equal to \(\log _{7} 5+\log _{7} 3 ?\) $$ \text { F. } \log _{7} 8 \quad \text { G. } \log _{7} 15 \quad \text { H. } \log _{7} 125 \quad 1 . \log _{49} 15 $$

Solve each equation. Check for extraneous solutions. \(\sqrt[3]{y^{4}}=16\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.