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Chapter 4: Problem 89

What question can be asked to help evaluate \(\log _{3} 81 ?\)

Short Answer

Expert verified
The log base 3 of 81 is 4. This can be deduced by re-writing the logarithm function in exponential form and comparing the exponents.

Step by step solution

01

Understand the Logarithm

The logarithm function \(\log _{a} b = n\) is equivalent to the expression \(a^n = b\). This means that \(n\) is the power to which \(a\) must be raised to get \(b\). In the provided problem, \(a = 3\) and \(b = 81\). We are to find \(n\). The aim is to evaluate \(\log _{3} 81\).
02

Rewrite in Exponential Form

Rewrite the logarithmic function in exponential form: \(3^n = 81\). We know that \(81\) is the same as \(3^4\) because \(3*3*3*3 = 81\). So we have \(3^n = 3^4\).
03

Compare the Exponents

The bases on both sides of the equation are equal (\(3 = 3\)), so the exponents should also be equal. That means \(n = 4\). This is possible due to the property that if \(a^b = a^c\) then \(b = c\) if \(a > 0\) and \(a 鈮� 1\).

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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?

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{16} 57,2 $$

Describe the product rule for logarithms and give an example.

Rescarch applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, consumption of natural resources, human memory, progress over time in a sport, profit over time. For the area that you select. explain how logarithmic functions are used and provide examples.
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