Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the logarithm. Round your result to three decimal places.\(\log _{9} \frac{3}{5}\)

Step-by-Step Solution

Answer

To find the final answer, divide the calculated logarithm of 3/5 by the logarithm of 9 and round it to three decimal places.

Step 1: Setup and Change of Base Formula

Express the logarithm using the change of base formula, \(\log_b a = \frac{\log_d a}{\log_d b}\), where d is any positive number not equal to 1. Hence, \(\log_9 \frac{3}{5} = \frac{\log \frac{3}{5}}{\log 9}\)

Step 2: Logarithm application

Calculate the value of numerator and denominator separately using logarithm. Logarithms can be calculated using tables or a digital calculator.

Step 3: Final calculation

Substitute the calculated values into the setup equation from step 1 and find the solution.

Key Concepts

Change of Base Formula

When dealing with logarithms, it's vital to know how to switch between bases. This is especially handy when your calculator does not directly support the base you're working with. The change of base formula allows this kind of flexibility. The formula is written as: \[\begin{equation} \log_b a = \frac{\log_d a}{\log_d b} \end{equation}\]where \( b \) and \( d \) are bases, \( a \) is the value we're taking the logarithm of, and \( d \) can be any positive number other than 1. Choosing \( d \) as 10 or \( e \) is common because most calculators have keys for \( \log \) (base 10, also known as common logarithm) and \( \ln \) (base \( e \), also known as natural logarithm).Using our exercise as an example:\[\begin{equation} \log_9 \frac{3}{5} = \frac{\log \frac{3}{5}}{\log 9} \end{equation}\]You simply divide the logarithm of the number by the logarithm of the new base. This is essentially converting a complex base into a more manageable one, facilitating further calculations.

Evaluating Logarithms

Evaluating logarithms may seem daunting, but once you understand the concept, it's just another mathematical operation. A logarithm questions, in essence, what power you must raise the base to, in order to get the number presented in the logarithm.For instance, if you see \( \log_9(3/5) \), you're being asked, 'To what power must I raise 9 to get 3/5?'There are a few methodologies to make the evaluation easier. One approach is to use logarithm properties such as the product, quotient, and power rules to simplify the expression before evaluating. If the base and the number are not simplified or if they are unfamiliar numbers, you're likely going to use a calculator.Remember when using a calculator:

• Make sure it's set to the correct mode (degree vs. radians, for example, if dealing with trigonometric functions).
• Understand whether it uses common or natural logarithms as default for the 'LOG' and 'LN' keys respectively.

In the aforementioned example, since 9 is not a base readily available on calculators, you'd use the change of base formula to evaluate the expression.

Logarithmic Calculations

Logarithmic calculations are essentially about understanding exponential relationships in a different way. After grasping how to change bases and evaluate a basic logarithm, combining these skills allows us to handle more complex calculations.To calculate the value, you can follow these steps:

1. Isolate the logarithm (if part of a larger expression).
2. If necessary, simplify the logarithm using properties or change the base.
3. Evaluate the simplified logarithm using a calculator or a logarithm table.
4. Round the result according to the necessary precision, as often seen in practical applications.

In the original exercise, the final step after applying the change of base formula and evaluating the simplified logarithm is to round the result to three decimal places to get the answer. This precision ensures that the answer is both practical and usable in real-life scenarios where exact values are often not required.