/*! 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 83 Determine whether the statement ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 83

Determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{6} x\) is a reflection of the graph of \(g(x)=6^{x}\) in the \(x\) -axis.

Short Answer

Expert verified
No, the statement is false. The graph of \(f(x) = \log_{6}x\) is not the reflection of the graph of \(g(x) = 6^{x}\) in the x-axis. Instead, the two graphs are inverses of each other.

Step by step solution

01

Understanding the functions

The general shape of an exponential function is increasing if the base is greater than 1, and decreasing if the base is between 0 and 1. The general shape of a logarithmic function is increasing for all positive values of \(x\). The graph of \(f(x) = \log_{6}x\) is the inverse of the graph of \(g(x) = 6^x\), but not its reflection.
02

Analysing the proposition

If \(\log_{6}x\) would be the reflection of \(6^{x}\), that would mean for every \(x\) such that \(f(x) = a\), we would have \(g(x) = -a\). However, this is not true. The graph of \(f(x) = \log_{6}x\) is the inverse of \(6^{x}\), meaning that for a value \(a\), \(f(x) = a\) would mean \(g(a) = x\), not \(g(x) = -a\). Therefore, the statement is not correct: The graph of \(f(x) = \log_{6}x\) is not the reflection of the graph of \(g(x) = 6^{x}\) across the x-axis.

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!

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

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{5} \frac{1}{125}$$

Condensing a Logarithmic Expression In Exercises \(67-82\) , condense the expression to the logarithm of a single quantity. $$\frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right]$$

Curve Fitting In Exercises \(89-92,\) find a logarithmic equation that relates \(y\) and \(x .\) Explain the steps used to find the equation. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline y & {1} & {1.587} & {2.080} & {2.520} & {2.924} & {3.302} \\\ \hline\end{array} $$

Condensing a Logarithmic Expression In Exerrises \(67-82\) , condense the expression to the logarithm of a single quantity. $$\frac{2}{3} \log _{7}(z-2)$$

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{5} 75-\log _{5} 3$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.