/*! 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 26 Simplify. $$ \log _{3} 3 $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 26

Simplify. $$ \log _{3} 3 $$

Short Answer

Expert verified
1

Step by step solution

01

Identify the logarithmic property

Identify which logarithmic property can be used to simplify the given expression. Here, the expression is \(\text{log}_{3} 3 \). One useful property is \[ \text{log}_{b} b^{x} = x \].
02

Apply the property

Using the property \(\text{log}_{b} b^{x} = x\), we can see that \(\text{log}_{3} 3\) simplifies directly to 1. This is because \(\text{log}_{3} 3\) is the same as asking the power to which 3 must be raised to get 3, which is 1.

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.

Logarithm
A logarithm answers the question: 'To what power must we raise a specific number to obtain another number?' This specific number is called the base of the logarithm. In mathematical terms, if you have \(a = b^c\), then \(c = \log_b(a)\). Here, \(c\) is the logarithm of \(a\) with base \(b\).

For example, \(\log_3(9) = 2\) because 3 raised to the power of 2 equals 9 (\(3^2 = 9\)). The logarithmic function is the inverse of exponentiation.
Base
The base of a logarithm is the number that is raised to a power to produce another number. In our example, the base is 3 in \(\log_3(3)\).

Understanding the base is crucial because it tells us what number we are repeatedly multiplying.

Some important points to remember about the base:
  • The base must be a positive number and cannot be 1.
  • Common bases include 10 (common logarithm) and e (natural logarithm).
  • Changing the base will change the value of the logarithm.


Knowing the base helps you decode what the logarithm represents.
Simplification
Simplification in logarithms often involves using well-known properties. One such property is \(\log_b(b^x) = x\).

In our exercise, we had \(\log_3(3)\). Following the property, we see this equals 1. Why? Because 3 raised to the power of 1 is 3, so \(\log_3(3) = 1\).

Simplifying can make calculations easier and allows you to work with more straightforward expressions. Other ways to simplify include:
  • Combining like terms.
  • Using the property \(\log(ab)=\log{a} + \log{b}\).
  • Applying the change of base formula.
Exponentiation
Exponentiation is the operation of raising one number, the base, to the power of another number, the exponent. For instance, in \(3^2 = 9\), 3 is the base and 2 is the exponent.

Logarithms and exponentiation are inverse operations. If you know \(3^2 = 9\), then the logarithm \(\log_3(9) = 2\) because 3 must be raised to the power of 2 to get 9.

Some key points about exponentiation:
  • Any number raised to the power of 0 is 1 (\(a^0 = 1\)).
  • Any number raised to the power of 1 is the number itself (\(a^1 = a\)).
  • Exponentiation follows specific rules, such as \(a^m \cdot a^n = a^{m+n}\).

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

The number of bacteria in a culture grows at an exponential growth rate of \(139 \%\) per hour. What is the doubling time for these bacteria?

Solve Explain why a logarithm base must be positive.

Solve. If no solution exists, state this. $$ 3^{2 x}-8 \cdot 3^{x}+15=0 $$

Find each of the following, given that $$ f(x)=\frac{1}{x+2} \quad \text { and } \quad g(x)=5 x-8 $$ $$ (f+g)(0) $$

Simplify. $$ \log _{81} 3 \cdot \log _{3} 81 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.