Problem 145 - College Algebra

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: Apply the change of base formula

We will change the base of both logarithms to 2, an arbitrary choice that will simplify comparison. To do that we use the change of base formula. \[ \log _{4} 60 = \frac{\log_2 {60}}{\log_2 {4}} \] and \[ \log _{3} 40 = \frac{\log_2 {40}}{\log_2 {3}} \].

Step 2: Simplify

Now simplifying these expressions, we get: \[ \log _{4} 60 = \frac{\log_2 {60}}{2} \] because \( \log_2 {4} \) equals 2, and \[ \log _{3} 40 = \frac{\log_2 {40}}{\log_2 {3}} \] can't be reduced further since \( \log_2 {3} \) is not a whole number.

Step 3: Compare the values

Now, by comparing these two values, it's clear that dividing by 2 produces a smaller value than dividing by \( \log_2 {3} \). Therefore, \( \log _{4} 60 \) is smaller than \( \log _{3} 40 \), which means \( \log _{3} 40 \) is greater.