Problem 86 - Algebra and Trigonometry

Question

Comparing Logarithms Which is larger, log \(_{4} 17\) or \(\log _{5} 24 ?\) Explain your reasoning.

Step-by-Step Solution

Answer

\( \log_{4} 17 \) is larger than \( \log_{5} 24 \).

Step 1: Use the Change of Base Formula

The first step to comparing \( \log_{4} 17 \) and \( \log_{5} 24 \) is to use the change of base formula for logarithms:\[\log_b a = \frac{\log_c a}{\log_c b}\]This allows us to express both logarithms with the same base, typically base 10 or natural logarithms (base \(e\)). Here we will use base 10 for simplicity.

Step 2: Compute Each Logarithm with Base 10

Using the change of base formula, we find:\[\log_{4} 17 = \frac{\log_{10} 17}{\log_{10} 4}\]\[\log_{5} 24 = \frac{\log_{10} 24}{\log_{10} 5}\]Calculate each value using a calculator.

Step 3: Evaluate the Quotients

Substitute the approximate values of the logs:- \( \log_{10} 17 \approx 1.2304 \)- \( \log_{10} 4 \approx 0.6021 \)- \( \log_{10} 24 \approx 1.3802 \)- \( \log_{10} 5 \approx 0.6989 \)Calculate each quotient:\[\log_{4} 17 = \frac{1.2304}{0.6021} \approx 2.043\]\[\log_{5} 24 = \frac{1.3802}{0.6989} \approx 1.975\]

Step 4: Compare the Results

We now compare the calculated values:\( \log_{4} 17 \approx 2.043 \) and \( \log_{5} 24 \approx 1.975 \).Clearly, \( \log_{4} 17 \) is larger than \( \log_{5} 24 \).

Key Concepts

Change of Base Formula

The change of base formula is a key concept in logarithms, allowing us to convert logarithms of any base into those with a base more convenient for calculations, often base 10. This is particularly helpful because most scientific calculators are set up to calculate logarithms in base 10 directly. The formula is given as:

• \( \log_b a = \frac{\log_c a}{\log_c b} \)

Here, \( b \) is the original base of the logarithm, and \( c \) is the base to which you'd like to change, commonly base 10 or the natural logarithm base \( e \).
This allows for simpler, more manageable calculations and makes it easier to compare logarithms with different bases by converting them to a common base.

Comparing Logarithmic Values

When comparing logarithmic values with different bases, it's crucial to express them in the same base using the change of base formula. Doing so simplifies the comparison process and provides a clear numerical basis to determine which value is larger.

• For example, converting \( \log_{4} 17 \) and \( \log_{5} 24 \) into base 10 logarithms involves using the formula to evaluate both expressions using base 10, after which you calculate the logarithm values with the help of a calculator.
• Once the base 10 equivalent values are obtained, it becomes straightforward to compare these: a higher logarithmic value signifies a larger logarithm.

In essence, this method transforms a potentially complex comparison into a simple arithmetic task of comparing decimal numbers.

Base 10 Logarithms

Base 10 logarithms, or common logarithms, are denoted as \( \log_{10} \,a \). They are widespread in both mathematical computations and real-world applications because they are convenient to use on standard calculators, allowing for quick calculations without additional conversion steps.

• When comparing or evaluating expressions that involve exponential growth or decay, base 10 logarithms often serve as a starting point for simplification.
• The simplicity of using base 10 makes it an ideal choice for many practical applications such as pH calculations in chemistry or measuring earthquake magnitudes.

In mathematical problems, converting logarithms to base 10 can aid in problem-solving by transforming complex logarithmic expressions into basic arithmetic operations, thus making tasks more approachable for students and professionals alike.