/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 109 Describe the change-of-base prop... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 109

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

Short Answer

Expert verified
The change-of-base property states that \( \log_b c = \frac{\log_a c}{\log_a b} \). An illustration is that to calculate \( \log_4 16 \), using a calculator that only has a base of 10, we can apply the change-of-base formula: \( \log_4 16 = \frac{\log_{10} 16}{\log_{10} 4} \), which results in \( 2 \).

Step by step solution

01

Describe the change-of-base property

The change-of-base property is an important logarithmic rule that states that for any positive number \( a \), \( b \), and \( c \), where \( a 鈮� 1 \) and \( b 鈮� 1 \), the equation \( \log_b c = \frac{\log_a c}{\log_a b} \) holds true. This property allows us to compute the logarithm of any number using any base of our choice.
02

Provide an example of the change-of base property

Let's consider an example to illustrate this property. Given that we need to calculate \( \log_4 16 \), but our calculator only has a base of 10, we can use the change-of-base formula to perform this calculation: \( \log_4 16 = \frac{\log_{10} 16}{\log_{10} 4} \) to get \( 2 \) as the result.

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Most popular questions from this chapter

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts \((a)-(c) .\) Try generalizing this observation.

What is the natural exponential function?

Without using a calculator, find the exact value of $$ \frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001} $$

The functions $$ f(x)=6.43(1.027)^{x} \quad \text { and } \quad g(x)=\frac{40.9}{1+6.6 e^{-0.049 x}} $$ model the percentage of college graduates, among people ages 25 and older, \(f(x)\) or \(g(x), x\) years after \(1950 .\) Use these functions to solve. (BAR GRAPH CAN'T COPY) Which function is a better model for the percentage who were college graduates in \(1990 ?\)

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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