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Chapter 12: Problem 119

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 \(\log _{3} 40\).

Step by step solution

01

Use Logarithmic Laws

Let's use the change-of-base formula to both expressions, converting them into natural logarithm form. The formula is \(\log_b a = \frac{{\ln a}}{{\ln b}}\), where \(\ln\) is the natural logarithm. Thus, \(\log _{4} 60\) becomes \(\frac{{\ln 60}}{{\ln 4}}\), and \(\log _{3} 40\) becomes \(\frac{{\ln 40}}{{\ln 3}}\).
02

Deduce the Relationship

Although numbers are not comparable directly, natural logarithms of integers increase the larger the integers. Therefore, comparing two fractions with the same denominator, the fraction with the larger numerator is bigger. Here, because 60 is greater than 40, and 4 is larger than 3, the denominator of \(\frac{{\ln 60}}{{\ln 4}}\) will increase faster than the numerator of \(\frac{{\ln 40}}{{\ln 3}}\). So, \(\frac{{\ln 60}}{{\ln 4}} < \frac{{\ln 40}}{{\ln 3}}\), meaning \(\log _{4} 60 < \log _{3} 40\).

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Most popular questions from this chapter

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

Describe the quotient rule for logarithms and give an example.

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Solve each equation in Exercises \(144-146 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

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. $$3^{x+1}=9$$
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