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Chapter 13: Problem 66

Evaluate each logarithm. $$\log _{3} \frac{1}{3}$$

Short Answer

Expert verified
The short answer is \(\log_{3} \frac{1}{3} = -1\), as the exponent to which the base must be raised to obtain \(\frac{1}{3}\) is -1.

Step by step solution

01

Analyze the logarithm

We are given the logarithm \(\log_{3} \frac{1}{3}\), and we need to find the value of the exponent x that satisfies the equation: \(3^x = \frac{1}{3}\). In other words, to what power should 3 be raised to obtain the value \(\frac{1}{3}\)?
02

Identify the inverse property of logarithms

One of the properties of logarithms states that: \(b^{\log_{b}(x)} = x\) Recalling this property, we want to rewrite \(\frac{1}{3}\) in a form that will help us identify the exponent and solve the problem.
03

Rewrite the fraction

Observe that \(\frac{1}{3}\) can also be written as \(3^{-1}\), since any number raised to the power of -1 is equal to its reciprocal. Hence, we have: \(3^x = 3^{-1}\)
04

Find the exponent

From step 3, we now have an equation in the form of \(3^x = 3^{-1}\). Since the bases are the same, we know that the exponents must be equal. Therefore, x must be equal to -1. So, \(\log_{3} \frac{1}{3} = -1\).

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Key Concepts

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

Inverse Property of Logarithms
Logarithms are the mathematical "undo" function for exponents. The inverse property is like a magical key that helps us solve logarithmic equations. Specifically, this property states that for any positive number \(b\), except 1, and any real number \(x\), the expression \(b^{\log_{b} (x)} = x\) holds true.

In simpler terms, if you take a number and apply a logarithm with the same base, and then raise it as a power, you end up right back where you started. This makes solving logarithmic equations straightforward when both sides share the same base.
Exponents
Exponents are a way to express repeated multiplication. When you see an expression like \(3^2\), it simply means 3 multiplied by itself once, or 9. Likewise, \(3^{-1}\) represents 1 divided by 3, which equals \(\frac{1}{3}\).

Recognizing and working with exponents is essential when dealing with logarithms. In our exercise, knowing that \(\frac{1}{3}\) equals \(3^{-1}\) allowed us to match the bases and find the solution quickly.
Base of Logarithm
The base of a logarithm tells us which number we are repeatedly multiplying in an exponential process. Consider \(\log_{3} x\): here, 3 is the base. When evaluating logarithms like \(\log_{3} \frac{1}{3}\), understanding the relationship between base and exponent is key.

This base also dictates the growth pattern of the function. For our exercise, since the base was 3, we focused on powers of 3 and their reciprocals. Always ensure the base remains consistent while applying logarithmic identities or solving equations.
Reciprocal
The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 3 is \(\frac{1}{3}\), and for \(\frac{1}{3}\), it is 3. Learning about reciprocals allows us to use negative exponents efficiently.

When we rewrite \(\frac{1}{3}\) as \(3^{-1}\), we're using the concept of reciprocals. This transformation simplifies logarithmic equations, as it often turns division into multiplication, thus aligning bases for easy comparison and solution.

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