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Chapter 4: Problem 145

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

Short Answer

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

Step by step solution

01

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 \(\frac{1}{1.585} \log_{2} 40\).
02

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 \(2^{40/1.585}\).
03

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

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

he formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 . a. Find Hungary's population, in millions, for \(2006,2007\), \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It's important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.

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

Find the domain of each logarithmic function. $$f(x)=\log \left(\frac{x-2}{x+5}\right)$$
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