Problem 88 - Elementary and Intermediate Algebra

Question

Assume that \(\log 4 \approx 0.6021, \log 7 \approx 0.8451,\) and \(\log 9 \approx 0.9542 .\) Use these values to evaluate each logarithm. \(\log _{b} \frac{7}{4}\)

Step-by-Step Solution

Answer

\(\log_b \frac{7}{4} \approx 0.243\)

Step 1: Identify the Given Logs

We are given approximate values for the logarithms: \( \log 4 \approx 0.6021 \), \( \log 7 \approx 0.8451 \), and \( \log 9 \approx 0.9542 \). We need to evaluate \( \log_b \frac{7}{4} \).

Step 2: Apply Logarithm Property

Recall the property of logarithms that states \( \log_b \frac{m}{n} = \log_b m - \log_b n \). This means we can express \( \log_b \frac{7}{4} \) as \( \log_b 7 - \log_b 4 \).

Step 3: Use Given Log Values

Substitute the given values into the expression from Step 2: \( \log_b \frac{7}{4} = \log_b 7 - \log_b 4 \approx 0.8451 - 0.6021 \).

Step 4: Perform the Subtraction

Calculate the difference: \( 0.8451 - 0.6021 = 0.243 \).

Step 5: Conclusion

The value of \( \log_b \frac{7}{4} \) is approximately 0.243.

Key Concepts

Properties of Logarithms

Logarithms can seem tricky at first glance, but understanding their properties can simplify many calculations. One of the fundamental properties is the quotient rule. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.This is expressed mathematically as:\[ \log_b \frac{m}{n} = \log_b m - \log_b n \]For our particular exercise, where we need to evaluate \( \log_b \frac{7}{4} \), the quotient property tells us that the expression can be rewritten as:\[ \log_b 7 - \log_b 4 \]This property is incredibly useful because it transforms complex division within the logarithm into simple subtraction, thus easing the calculation process. Understanding and applying the properties of logarithms like this one can greatly assist in handling more complex logarithmic expressions.

Logarithmic Expressions

Logarithmic expressions often involve transforming a numerical relationship into one that is easier to manage. A logarithmic expression might involve base changes or simplifications using known logarithm values.For students, it is important to remember the concept that

• logarithms are the inverse operations of exponentials
• they help in solving equations where the variable is an exponent

In our exercise, simplifying \( \log_b \frac{7}{4} \) into smaller parts using the values of \( \log 7 \) and \( \log 4 \) is an effective approach. With the given approximate logarithmic values, students can substitute these directly into the expression, making explicit calculations feasible and straightforward.The power of these expressions lies in their ability to reduce complex operations into manageable parts. This especially comes in handy when calculators or direct measurements are unavailable, such as in theoretical math problems or exams.

Logarithmic Evaluation

Evaluating logarithms is about finding the numerical value of a logarithmic expression. In scenarios where exact calculations are crucial, known values can guide the process.In our exercise, we utilize known logarithmic values in the evaluation of \( \log_b \frac{7}{4} \). We substitute the approximate values:

• \( \log 4 \approx 0.6021 \)
• \( \log 7 \approx 0.8451 \)

Then, applying the properties of logarithms leads us to conduct a simple subtraction:\[ \log_b \frac{7}{4} = \log_b 7 - \log_b 4 \approx 0.8451 - 0.6021 \]Continuing with the subtraction gives a result of approximately 0.243. Accurate evaluation like this is essential to ensure solutions are reliable, making understanding and application of logarithms a vital mathematical skill.