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Chapter 4: Problem 111

You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.

Short Answer

Expert verified
The property of logarithms in question is the Quotient Rule of Logarithms, which states that the logarithm of a quotient is the difference of the logarithms. It means division inside the log can be translated to subtraction outside the log, such as \( \log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\).

Step by step solution

01

Introduction to Logarithmic Property

The logarithmic property in question is often referred to as the Quotient Rule of Logarithms. It is one of the three basic properties of logarithms, together with the Product Rule and the Power Rule, which allow you to simplify expressions or solve equations involving logarithms.
02

The Quotient Rule of Logarithms Explained

The Quotient Rule states that the logarithm of a quotient is the difference of the logarithms. In mathematical terms, it can be expressed as follows: If you have two positive real numbers, \(a\) and \(b\), and \(a \neq 1\), then for every \(x > 0\) and \(y > 0\), the rule is as such: \( \log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\) . This means, when you are dividing within a log expression, you can convert it to subtraction outside of the log.
03

Example

For example, suppose we have \( \log_2\left(\frac{8}{4}\right)\). Using the Quotient Rule of Logarithms, this division inside the logarithm can be rewritten as subtraction outside the logarithm, like so: \( \log_2\left(\frac{8}{4}\right) = \log_2(8) - \log_2(4)\). After calculating the log values, it becomes \(3 - 2 = 1\), which is indeed the same as \( \log_2(2)\).

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Most popular questions from this chapter

In Exercises 139鈥�142, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4} $$

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in 2013 , Uganda's population will exceed Canada's.

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

Explaining the Concepts Describe a difference between exponential growth and logistic growth.
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