/*! 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 8 Decide whether cach statement is... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 8

Decide whether cach statement is true or false. $$\log _{3} 4^{5}=5 \log _{3} 4$$

Short Answer

Expert verified
The given statement is \(\log _{3} 4^{5}=5 \log _{3} 4\). By applying the Power Rule for logarithms, we obtain \(5 \log_{3} (4) = 5 \log_{3} (4)\). Since both sides of the equation are equal, the statement is true.

Step by step solution

01

Identify the given statement

The given statement is: \(\log _{3} 4^{5}=5 \log _{3} 4\)
02

Apply the Power Rule to the left side of the equation

We apply the Power Rule to the left side of the equation, where the base is \(3\), exponent is \(5\), and the positive number is \(4\): \(\log_{3} (4^5) = 5 \log_{3} (4)\)
03

Compare the expressions

Now we compare the resulting expression with the right-hand side of the given equation: \(5 \log_{3} (4) = 5 \log_{3} (4)\)
04

Decide whether the statement is true or false

Since both sides of the equation are equal: \(5 \log_{3} (4) = 5 \log_{3} (4)\) We can conclude that the given statement is true.

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.

Understanding the Power Rule
The Power Rule for logarithms is a useful algebraic tool. It helps in breaking down the exponent within a logarithm expression. When you have a logarithm with an exponent, like \( \log_b(a^n) \), you can simplify it by bringing the exponent in front of the logarithm. This transforms it into \( n \log_b(a) \). This is particularly helpful when solving equations, making complex expressions simpler to handle.
  • Example: If you have \( \log_{3}(4^5) \), applying the Power Rule gives you \( 5 \log_{3}(4) \).
  • Benefit: This rule allows for easier manipulation and understanding of logarithmic expressions.
What is Exponentiation?
Exponentiation is a fundamental concept in mathematics. It involves raising a number, known as the base, to the power of an exponent. The base is multiplied by itself, as many times as the exponent dictates. For example, \( 4^5 \) means 4 is multiplied by itself five times. The result is a much larger number, which logarithms can help manage.
  • Key Point: The operation compresses repeated multiplication into a concise form.
  • Link to Logarithms: Logarithms can be used to reverse the process of exponentiation, by identifying how many times a base must be multiplied to reach a certain number.
Exploring the Base of Logarithm
The base of a logarithm is crucial in determining the scale of measurement. In \( \log_b(a) \), 'b' is the base. It determines the number of factors needed to reach 'a'. For example, in \( \log_{3}4 \), the question is: \"To what power must 3 be raised, to get 4?\" The base is always a positive number greater than 1, influencing the logarithm's properties significantly.
  • Common Bases: 10 (common logarithm), \( e \) (natural logarithm), 2 (binary logarithm).
  • Importance: The choice of base can affect the outcome and interpretation of a logarithmic calculation.
Algebra and Logarithms
Algebra is the language in which we express and manipulate mathematical ideas, including logarithms. Logarithmic functions can often appear in algebraic equations and need careful handling. Mastering algebra skills helps in rearranging and solving these expressions. You use rules like the Power Rule to simplify expressions, making them more manageable.
  • Critical Skills: Understanding expressions, solving equations, and manipulating formulas.
  • Link to Logarithms: Algebra provides the framework to solve complex logarithmic equations.

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

Determine whether each statement is true or false. If it is false, rewrite the statement so that it is true. If \(f^{-1}\) is the inverse of \(f\), then \(\left(f^{-1} \circ f\right)(x)=x\) and \(\left(f \circ f^{-1}\right)(x)=x\)

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log _{7} d-\log _{7} 3$$

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$4 \log _{3} t-2 \log _{3} 6-2 \log _{3} u$$

Do all functions have inverses? Explain your answer.

Find the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$h(x)=x+3$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.