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Chapter 12: Problem 99

Describe the change-of-base property and give an example.

Short Answer

Expert verified
The Change-of-Base property in logarithms allows logarithm base alterations. It is represented as \(\log_c b = \frac{\log_a b}{\log_a c}\). An example is \(\log_4 8 = \frac{\log_2 8}{\log_2 4} = 1.5\).

Step by step solution

01

Defining the Change-of-Base Property

The Change-of-Base property states that for any positive numbers \(a\), \(b\), and \(c\) where \(a \neq 1\) and \(b \neq 1\), the quotient of the logarithm of \(b\) with base \(a\) and the logarithm of \(c\) with base \(a\) is equal to the logarithm of \(b\) with base \(c\). Mathematically, it can be stated as: \[\log_c b = \frac{\log_a b}{\log_a c}\]
02

Giving an Example

Let's illustrate the Change-of-Base property with an example. Consider \(a = 2\), \(b = 8\), and \(c = 4\). The aim is to calculate \( \log_4 8 \). Utilizing the Change-of-Base property: \[ \log_4 8 = \frac{\log_2 8}{\log_2 4} = \frac{3}{2} = 1.5 \]

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

Solve: $$\frac{3}{x+1}-\frac{5}{x}=\frac{19}{x^{2}+x}$$

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a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$

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