/*! 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 34 Evaluate each expression without... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 34

Evaluate each expression without using a calculator. $$\log _{81} 9$$

Short Answer

Expert verified
The result of \(\log _{81} 9\) is 0.5

Step by step solution

01

Recognize the Base and the Argument

It's key to note that both 9 and 81 are powers of 3. \(81 = 3^{4}\) and \(9 = 3^{2}\), this enables us to express the expression in a more familiar base for simplification.
02

Employ the Change of Base Formula

The Change of Base Formula states that for \(log _{b} a\), it can be expressed as \(\frac{log _{c} a}{log _{c} b}\) where c is often 10 in a calculator or in cases like this, a number where the result can be known readily, in this case, 3. Hence \(log _{81} 9\) can be expressed as \(\frac{log _{3} 9}{log _{3} 81}\).
03

Deduce the Value

Given that \(9 = 3^{2}\) and \(81 = 3^{4}\), it applies that \(log _{3} 9 = 2\) and \(log _{3} 81 = 4\). Thus, \(\frac{log _{3} 9}{log _{3} 81}\) equals \(\frac{2}{4} = 0.5\)

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Most popular questions from this chapter

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(50 \% ?\)

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

Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 80 -year-olds have some coronary heart disease?

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 2350 dollar Annual Interest Rate 15.7% Accumulated Amount Triple the amount invested Time \(t\) in Years _______
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