Problem 119 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The greater number is \(\log _{3} 40\).

Step 1: 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 \(\frac{{\ln 40}}{{\ln 3}}\).

Step 2: 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 \(\log _{4} 60 < \log _{3} 40\).