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Chapter 4: Problem 5

Evaluate the logarithms. a. \(\log _{3} 81\) b. \(\log _{3} 3^{4}\)

Short Answer

Expert verified
Both \(\log _{3} 81\) and \(\log _{3} 3^{4}\) evaluate to 4.

Step by step solution

01

Understand the Logarithm Definition

Recall that \(\log_b(a) = c\) if and only if \(b^c = a\). This definition will help evaluate the logarithms.
02

Evaluate \(\log_{3} 81\)

Express 81 as a power of 3. Note that 81 = 3^4. Using the definition of logarithms, \(\log_{3} 81 = \log_{3} (3^4)\).
03

Simplify Using Logarithm Properties

Apply the property \(\log_b(b^k) = k\) to get \(\log_{3}(3^4) = 4\). Thus, \(\log_{3} 81 = 4\)
04

Evaluate \(\log_{3} 3^4\)

Using the property \(\log_b(b^k) = k\), it follows directly that \(\log_{3} 3^4 = 4\)
05

Conclusion

Both logarithms evaluate to the same value, namely 4.

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Key Concepts

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

Understanding Logarithm Definition
To get started with logarithms, it's crucial to understand what a logarithm really is. The logarithm \(\text{log}_b(a)\) answers the question: 'To what exponent must we raise \(b\) to get \(a\)?' In other words:
\[ \text{log}_b(a) = c \text{ if and only if } b^c = a \]
For example, \( \text{log}_2(8) = 3 \) because raising 2 to the power of 3 gives 8. Understanding this definition is the foundation for evaluating and working with logarithms.
Exploring Logarithm Properties
Logarithms have several properties that can simplify calculations:
  • Product Property: \( \text{log}_b(MN) = \text{log}_b(M) + \text{log}_b(N) \)
  • Quotient Property: \( \text{log}_b(\frac{M}{N}) = \text{log}_b(M) - \text{log}_b(N) \)
  • Power Property: \( \text{log}_b(M^k) = k \text{log}_b(M) \)
For the given problem, the Power Property is the most useful, particularly the rule that \( \text{log}_b(b^k) = k \). This means if you鈥檙e taking the log of a base raised to an exponent, the log equals the exponent directly.
Evaluating Logarithms
Let's evaluate the logarithms step-by-step:
Example a: \( \text{log}_3(81) \) - First, express 81 as a power of 3. Since 81 can be written as \( 3^4 \), the logarithm becomes \( \text{log}_3(3^4) \). Using the Power Property, simplify it directly to \( 4 \).
Example b: \( \text{log}_3(3^4) \) - This one is more straightforward. Apply the Power Property directly to get \( 4 \).
The answer in both cases is \( 4 \). This shows how understanding properties can simplify evaluating logarithms tremendously.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \ln 10=\frac{1}{\log e} $$

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log (x y)=(\log x)(\log y) $$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{4}(3 w+11)=\log _{4}(3-w)\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{6}(7 x-2)=1+\log _{6}(x+5)\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(801=23^{y}+6\)
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