Problem 145 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

\(\log _{4} 60\) is greater than \(\log _{3} 40\).

Step 1: Converting Base to a Common Base.

Let's rewrite the given expressions \(\log _{4} 60\) and \(\log _{3} 40\) to the same base of 2 to make comparison easier. We know that \(4 = 2^2\), hence we can write \(\log _{4} 60\) as \(\log _{2^2} 60\), and \(\log _{3} 40\) as \(\log_{2^{1.585}} 40\) (since \(3 \approx 2^{1.585}\)). Using the rule mentioned in the analysis, these become \(\frac{1}{2} \log_{2} 60\) and \(\frac{1}{1.585} \log_{2} 40\).

Step 2: Converting Logarithms to Exponentials.

Next, convert the logarithms to exponentials. The exponential of log base 2 of some number x is equal to 2 power x. Hence, the two transformed expressions from Step 1 can be written as \(2^{60/2}\) and \(2^{40/1.585}\).

Step 3: Comparing the Expressions.

Finally, we can compare the two transformed expressions. It's obvious that 2 power 30 is greater than 2 power approximately 25.5. Thus, we can say that \(\log _{4} 60 > \(\log _{3} 40\).