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Chapter 4: Problem 145

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\)

Short Answer

Expert verified
The greater number is \( \log_{4}60 \).

Step by step solution

01

Understanding the Task

The task is to compare two logarithms: \( \log_{4}60 \) and \( \log_{3}40 \) without using a calculator. This will be accomplished by making use of logarithm properties to express the two given logarithms in a comparable way.
02

Making use of properties of logarithms

Starting with \( \log_{4}60 \), we can write 60 as \( 4^{x} \). With base \( b = 4 \), and result \( 60 \), we have \( x = \log_{4}{60} \). This implies that the 4 raised to the power of \( x \) equals 60. Similarly for \( \log_{3}40 \), we can write 40 as \( 3^{y} \). This implies that 3 raised to the power of \( y \) equals 40.
03

Making a comparison

From the previous step, we gather that \( x \) and \( y \) represent the powers to which 4 and 3 must be raised to achieve 60 and 40 respectively. Hence, comparing \( x \) and \( y \), we find that \( X > Y \). This is because 4 requires a smaller power to reach 60 than 3 does to reach 40. Therefore, \( \log_{4}60 \) > \( \log_{3}40 \).

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Most popular questions from this chapter

Describe the change-of-base property and give an example.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ \log (x-15)+\log x=2 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.

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

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 power regression option to obtain a model of the form \(y=a x^{b}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?
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