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Chapter 12: Problem 82

Use the special properties of logarithms to evaluate each expression. \(\log _{3} 27\)

Short Answer

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Step by step solution

01

Recall the Definition of a Logarithm

A logarithm \(\log_b(x)\) answers the question: 'To what power must the base \(b\) be raised, to obtain the number \(x\)?'
02

Identify the Base and the Argument

In the given expression \(\log_{3} 27\), the base \(b\) is \(3\) and the argument \(x\) is \(27\). This means we need to find the power to which \(3\) must be raised to get \(27\).
03

Express the Argument as a Power of the Base

Write \(27\) as a power of \(3\). We know that \(27 = 3^3\).
04

Use Logarithm Property

According to the logarithm property, \(\log_b(b^y) = y\). Here, \(b = 3\) and \(y = 3\).
05

Evaluate the Logarithm

So, \(\log_{3}(3^3) = 3\). Hence, \(\log_{3}(27) = 3\).

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Key Concepts

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

Base and Argument
Understanding the base and argument is crucial when working with logarithms. The base of a logarithm is the number that is raised to a power, while the argument is the result of that exponentiation. In the expression \(\text{log}_b(x)\), \b\ is the base, and \x\ is the argument.
Let's take a closer look at the example, \(\text{log}_{3}(27)\). Here:
  • Base \b\ is 3,
  • Argument \x\ is 27

This expression asks us to find the power to which 3 must be raised to obtain 27. The base provides a foundation, while the argument represents what we get when we raise the base to some power.
Logarithm Properties
Logarithms follow specific properties that simplify calculations. One fundamental property used in our example is:
\(\text{log}_b(b^y) = y\)
This property essentially tells us that a logarithm of a base raised to some power returns that power. For \(\text{log}_{3}(27)\), we first express 27 as a power of 3:
\27 = 3^3\
Utilizing the property, we then rewrite the log expression:
\(\text{log}_{3}(27) = \text{log}_{3}(3^3)\)
According to the property, this results in just the exponent:
\(\text{log}_{3}(3^3) = 3\)
Evaluating Logarithms
Evaluating logarithms involves applying the properties and understanding the relationship between the base and the argument.
Let's summarize the steps for evaluating the expression \(\text{log}_{3}(27)\):
  • Identify the base and argument: here base is 3, and argument is 27.
  • Express the argument as a power of the base: \27 = 3^3\.
  • Apply the logarithm property: \(\text{log}_3(3^3) = 3\).
Therefore, \(\text{log}_{3}(27) = 3\).
By following these steps, you ensure a straightforward path to arriving at the correct answer. The key lies in recognizing how the base and its powers relate to the argument and leveraging that relationship via logarithm properties.

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Most popular questions from this chapter

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{3} \sqrt{2}\)

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{5} \frac{8}{3}$$

Determine whether each statement is true or false. $$\log _{6} 60-\log _{6} 10=1$$

Solve each equation. \(\log _{x} 9=\frac{1}{2}\)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{-0.103 x}=7$$
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