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Chapter 5: Problem 6

In Exercises \(5-10\) evaluate the expression without using a calculator. $$\log _{3} 81$$

Short Answer

Expert verified
The value of the logarithmic expression \(\log_{3} 81\) is 4.

Step by step solution

01

Understand the Problem

The expression \(\log _{3} 81\) asks, '3 to what power gives 81?'. This is the question that you need to answer.
02

Use Knowledge of Exponents

You know that \(3^{4} = 81\), as \(3 * 3 * 3 * 3 = 81\). This implies that 4 is the exponent to which 3 must be raised to give 81.
03

Make the Connection to Logarithms

Knowing that \(3^{4} = 81\), you can conclude that \(\log _{3} 81 = 4\), because 4 is the exact power that you raise 3 to get 81.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
In mathematics, exponents represent the number of times a number is multiplied by itself. For example, when we write \(3^4\), it means \(3\) is multiplied by itself \(4\) times: \(3 \times 3 \times 3 \times 3 = 81\). This process of using exponents is a shorthand way of expressing repeated multiplication.
  • Exponentiation consists of a base and an exponent. The base is the number being multiplied, while the exponent indicates how many times the base is used as a factor.
  • If the exponent is 2, the expression is also known as squaring; if the exponent is 3, it is known as cubing.
  • Understanding the role of exponents is essential in deciphering other mathematical concepts, such as logarithms.
Base
In mathematical expressions involving exponents and logarithms, the base is a critical component. It is the number that is raised to a power, as in the expression \(3^4\) where \(3\) is the base.
  • The base tells us which number is being multiplied repeatedly. For example, in \(5^2\), the number \(5\) is the base and it is multiplied by itself, resulting in \(5 \times 5 = 25\).
  • In logarithmic functions, the base is the number that we repeatedly multiply to get another number. For example, in \(\log_3 81\), the base is \(3\).
  • Commonly used bases include \(10\), known as the common logarithm, and \(e\), which is the base for natural logarithms.
Understanding the base of an exponent or a logarithm is crucial because it helps determine the relationship between the numbers involved.
Logarithmic Function
A logarithmic function is the inverse of an exponential function. If exponentiation answers 'repeated multiplication', logarithms answer the question, 'what power?' In its simplest form, a logarithm tells you what exponent you need to get a certain number from a specific base.
  • For example, the expression \(\log_3 81\) is asking, 'To what power must \(3\) be raised to yield \(81\)?'
  • We know that \(3^4 = 81\), so \(\log_3 81 = 4\). This result means that \(3\) raised to the power of \(4\) equals \(81\).
  • Logarithms can be used to solve exponential equations and are very useful in various fields like science, engineering, and finance for dealing with large numbers or exponential growth.
The logarithmic function is very helpful in simplifying complex calculations involving exponential growth and decay.

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