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Chapter 4: Problem 15

Evaluate the expression. (a) \(\log _{3} 3\) (b) \(\log _{3} 1\) (c) \(\log _{3} 3^{2}\)

Short Answer

Expert verified
(a) 1, (b) 0, (c) 2

Step by step solution

01

Understanding Logarithms

To evaluate a logarithm, such as \( \log_{3} b \), we need to find the power to which the base (3) must be raised to get the number \( b \).
02

Evaluating \( \log_{3} 3 \)

Since \( 3^{1} = 3 \), \( \log_{3} 3 = 1 \) because 3 must be raised to the power of 1 to result in 3.
03

Evaluating \( \log_{3} 1 \)

Any non-zero number raised to the power of 0 is 1. Therefore, \( 3^{0} = 1 \), so \( \log_{3} 1 = 0 \).
04

Evaluating \( \log_{3} 3^{2} \)

The expression \( \log_{3} 3^{2} \) can be rewritten as \( 2 \log_{3} 3 \). Since we found earlier that \( \log_{3} 3 = 1 \), it follows that \( 2 \log_{3} 3 = 2 \times 1 = 2 \).

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Key Concepts

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

Evaluating Logarithms
Logarithms help us understand the concept of exponents in reverse. To evaluate a logarithm like \( \log_{3} b \), we are essentially asking: "To what power must 3 be raised, so that the result is \( b \) ?" This inverse relationship makes it a powerful tool in mathematics. For example, in the exercise given:
  • \( \log_{3} 3 \) equals 1 because you need to raise 3 to the power of 1 to get 3. Indeed, \( 3^{1} = 3 \).
  • For \( \log_{3} 1 \), the result is 0 because 3 raised to the power of 0 gives you 1: \( 3^{0} = 1 \).
  • In the case of \( \log_{3} 3^{2} \), this simplifies to 2 because 3 to the power of 2 gives you 9, but since we have 3 squared within a logarithm, it results in \( 2 \log_{3} 3 \) which calculates to 2.
Evaluating logarithms is like solving a little puzzle and, with practice, it becomes a straightforward task.
Logarithmic Properties
Logarithms possess several unique properties that greatly simplify complex calculations. When dealing with expressions involving multiple logarithmic elements, these properties are essential:
  • Product Property: The logarithm of a product can be expressed as the sum of the logarithms of the individual factors: \( \log_b(mn) = \log_b m + \log_b n \).
  • Quotient Property: The logarithm of a quotient is the difference of the logarithms: \( \log_b(\frac{m}{n}) = \log_b m - \log_b n \).
  • Power Property: When an exponent is present, such as in \( \log_b (m^n) \), it can be simplified by bringing the exponent in front: \( n \log_b m \). This was used in evaluating \( \log_3 3^2 \), transforming it into \( 2 \log_3 3 \).
These properties simplify calculations and make it easier to work with logarithmic expressions and equations.
Logarithmic Functions
Logarithmic functions are fundamental in mathematics, representing the concept of logarithms as a function. They are the opposite of exponential functions. While exponential functions are of the form \( y = b^x \), logarithmic functions are \( y = \log_b x \). This inverse relationship is crucial in various scientific and engineering contexts.The graph of a logarithmic function \( y = \log_b x \) has distinct characteristics:
  • The graph passes through the point (1,0) because \( \log_b 1 = 0 \) for any base \( b \).
  • It approaches negative infinity as \( x \) approaches 0 from the right but never touches the y-axis, showing it is undefined for non-positive values of \( x \). This makes the x-axis an asymptote.
  • The growth is gradual, indicating that logarithmic functions increase more slowly compared to linear or polynomial functions.
Understanding these functions helps in grasping their real-world applications, ranging from calculating the pH in chemistry to understanding sound intensity in decibels.

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Most popular questions from this chapter

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True or False? Discuss each equation and determine whether it is true for all possible values of the variables. (Ignore values of the variables for which any term is undefined.) (a) \(\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}\) (b) \(\log _{2}(x-y)=\log _{2} x-\log _{2} y\) (c) \(\log _{5}\left(\frac{a}{b^{2}}\right)=\log _{5} a-2 \log _{5} b\) (d) \(\log 2^{x}=z \log 2\) (e) \((\log P)(\log Q)=\log P+\log Q\) (f) \(\frac{\log a}{\log b}=\log a-\log b\) (g) \(\left(\log _{2} 7\right)^{x}=x \log _{2} 7\) (h) \(\log _{a} a^{a}=a\) (i) \(\log (x-y)=\frac{\log x}{\log y}\) (j) \(-\ln \left(\frac{1}{A}\right)=\ln A\)
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