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Chapter 4: Problem 86

Describe the quotient rule for logarithms and give an example.

Short Answer

Expert verified
The quotient rule for logarithms states that the logarithm of a quotient equals the difference of the logarithms of the numerator and denominator. An example is: log base 2 of 2, calculated from log base 2 of 8 minus log base 2 of 4, equals 1.

Step by step solution

01

Describe the Quotient Rule for Logarithms

The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. This can be written mathematically as: \[ \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \] where b is the base of the logarithm, and m and n are any two positive real numbers distinct from 1.
02

Give an Example

To understand this rule better, consider an example. Let \(b = 2\), \(m = 8\), and \(n = 4\). Substituting these values into the formula, the equation becomes: \[ \log_2(\frac{8}{4}) = \log_2(8) - \log_2(4) \]. On the left side, \( \frac{8}{4} = 2\), so \( \log_2(2) = 1 \). On the right side, using base 2, \( \log_2(8) = 3 \) and \( \log_2(4) = 2\). So, \( 3 - 2 = 1 \). Both sides of the equation yield the same result, proving the quotient rule for this example.

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Most popular questions from this chapter

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human car. The formula $$ D=10\left(\log I-\log I_{0}\right) $$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter". and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{\sqrt[3]{x} y^{4}}{z^{5}}\right) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(x^{2} y\right) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{4}\left(\frac{64}{y}\right) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{4}\left(\frac{\sqrt{x}}{64}\right) $$
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