/*! 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 89 Explain why \(\log _{3} 100\) is... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 89

Explain why \(\log _{3} 100\) is between 4 and 5 .

Short Answer

Expert verified
Since \(100\) is greater than \(81=3^4\) and less than \(243=3^5\), the exponent to which 3 must be raised to obtain 100 is greater than 4 and less than 5. Therefore, \(4<\log_3 100<5\), so \(\log_3 100\) is between 4 and 5.

Step by step solution

01

Find \(\log_3 81\) and \(\log_3 243\)

To find the value of \(\log_3 81\) and \(\log_3 243\), recall that the logarithm is the exponent to which the base must be raised to obtain the given number. In this case, the base is 3, so we are looking for the exponent to which 3 must be raised to obtain 81 and 243, respectively. Given that \(81 = 3^4\) and \(243 = 3^5\), we can conclude that: \[\log_3 81 = 4\] and \[\log_3 243 = 5\]
02

Compare \(\log_3 100\) with \(\log_3 81\) and \(\log_3 243\)

Now, we want to show that \(\log_3 100\) is between \(\log_3 81\) and \(\log_3 243\). To do this, we need to compare the values of 100 with 81 and 243. We already know that \(3^4 < 100 < 3^5\), since \(3^4 = 81\) and \(3^5 = 243\). Since \(100\) is greater than \(81\), it means that the exponent to which 3 must be raised to obtain 100 must be greater than the exponent to which 3 must be raised to obtain 81 (which is 4). Conversely, as \(100\) is less than \(243\), it means the exponent to which 3 must be raised to obtain 100 must be less than the exponent to which 3 must be raised to obtain 243 (which is 5).
03

Conclude that \(\log_3 100\) is between 4 and 5

We have established that the exponent to which 3 must be raised to obtain 100 is greater than 4 and less than 5. Therefore, we conclude that: \[4<\log_3 100<5\] So, \(\log_3 100\) is between 4 and 5.

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Most popular questions from this chapter

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=4-2 e^{8 x}\)

(a) Show that $$ 1.01^{100}

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