/*! 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 67 Use the special properties of lo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 67

Use the special properties of logarithms to evaluate each expression. \(\log _{3} 3\)

Short Answer

Expert verified
\( \log_{3} 3 = 1 \)

Step by step solution

01

Identify the Base and Argument

The given expression is \( \log_{3} 3 \). Here, the base of the logarithm is 3, and the argument is also 3.
02

Use the Logarithm Property

Recall the property of logarithms that states \( \log_{b} b = 1 \). This property holds because any number raised to the power of 1 equals itself.
03

Apply the Property

Using the property mentioned, \( \log_{3} 3 = 1 \).

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.

Logarithms
A logarithm answers the question: to what exponent must the base be raised, to produce a given number? For example, in the logarithm \(\text{log}_{b}a\), 'b' is the base and 'a' is the argument. The result of the logarithm is the exponent 'x' such that \({b^x = a}\).
When we encounter \(\text{log}_{3} 3\), it is asking: '3 raised to what power equals 3?'
  • In this scenario, the exponent is 1 because \({3^1 = 3}\).
  • Therefore, \(\text{log}_{3} 3 = 1\).
Logarithms are incredibly useful in solving equations with exponents, understanding growth patterns, and performing calculations in various fields like science and engineering.
Base and Argument
In a logarithmic expression, the two main components we need to focus on are the base and the argument.
Here鈥檚 a simple way to remember them:
  • The 'base' is the number that is being raised to a power. It is written as a subscript.
  • The 'argument' is the number we are trying to represent as a power of the base. It is written next to the base but not as a subscript.
For example, in \(\text{log}_{3} 3\):
  • The 'base' is 3.
  • The 'argument' is also 3.
This tells us that we need to find the exponent to which 3 must be raised to result in 3. In this case, the answer is 1, because \({3^1 = 3}\).
Understanding the roles of the base and argument helps in simplifying and solving logarithmic expressions accurately.
Logarithm Property
There are several properties of logarithms that make it easier to work with them. One such common property is:
\(\text{log}_{b} b = 1\).
  • This property states that any number raised to the power of 1 is the number itself.
  • In general, \(\text{log}_{b} b = 1\), for any base 'b'.

Applying this property to our example: In \(\text{log}_{3} 3\), the base is 3 and the argument is also 3. By using the property \(\text{log}_{b} b = 1\):
  • We know that \(\text{log}_{3} 3 = 1\)
. Recognizing and utilizing this property simplifies solving logarithmic equations. It is especially useful in quick evaluations and calculations, saving both time and effort.

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

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

Write in logarithmic form. \(7^{0}=1\)

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{19} 0.8325\)

Find the amount of money in an account after 12 yr if \(\$ 5000\) is deposited at \(7 \%\) annual interest compounded as follows. (a) Annually (b) Semiannually (c) Quarterly (d) Daily (Use \(n=365 .\) ) (e) Continuously

Solve each equation. Give exact solutions. \(\log _{5}(3 t+2)-\log _{5} t=\log _{5} 4\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.