Problem 145 - Algebra and Trigonometry

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 \( \log_{4}60 \).

Step 1: 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.

Step 2: 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.

Step 3: 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, \( \log_{4}60 \) > \( \log_{3}40 \).