Problem 16 - Algebra 2

Question

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} \frac{3}{2}\)

Step-by-Step Solution

Answer

\( \log _{5} \frac{3}{2} \approx 0.2519 \)

Step 1: Identify the Logarithmic Rule

We need to find the value of \( \log _{5} \frac{3}{2} \). Recognize that this expression can be simplified using the logarithmic property for division: \( \log_{b} \left( \frac{x}{y} \right) = \log_{b}(x) - \log_{b}(y) \).

Step 2: Apply the Logarithm Property

Using the rule \( \log_{b} \left( \frac{x}{y} \right) = \log_{b}(x) - \log_{b}(y) \), rewrite \( \log _{5} \frac{3}{2} \) as \( \log _{5} 3 - \log _{5} 2 \).

Step 3: Substitute Known Values

Substitute the given approximations into the rewritten expression: \( \log _{5} 3 - \log _{5} 2 \approx 0.6826 - 0.4307 \).

Step 4: Calculate the Result

Perform the subtraction: \( 0.6826 - 0.4307 = 0.2519 \). Thus, \( \log _{5} \frac{3}{2} \approx 0.2519 \).

Key Concepts

Division Property of Logarithms

The division property of logarithms is a fundamental and convenient tool in simplifying logarithmic expressions. When we have a division within the argument of a logarithm, we can separate it into the subtraction of two individual logarithms. Mathematically, this is represented as:

• \( \log_{b} \left( \frac{x}{y} \right) = \log_{b}(x) - \log_{b}(y) \)

Understanding this property can significantly simplify complex logarithmic problems and is crucial when working with various bases.
In our exercise, this allowed us to transform \( \log_{5} \left( \frac{3}{2} \right) \) into \( \log_{5}(3) - \log_{5}(2) \). This shift effectively breaks down one complex expression into simpler, separate parts that are easier to handle.

Approximate Logarithms

Approximating logarithms means finding a value that is close to the actual logarithmic value of the expression. Exact calculations, especially for non-standard bases like 5 in our exercise, may not always be feasible without computational tools.
This is where approximations come to the rescue, allowing us to perform calculations with known values to achieve a practical and useful result. In our scenario, we use the given approximations:

• \( \log_{5} 2 \approx 0.4307 \)
• \( \log_{5} 3 \approx 0.6826 \)

Using these approximations, we substitute them into our expression to get an approximate solution for \( \log_{5} \frac{3}{2} \). Approximations help make solvable what would otherwise require a calculator or more complex software.

Calculating Logarithms

Calculating logarithms often involves using known values and properties to simplify problems. In our exercise, we calculated the logarithm of a fraction using the division property and known approximations. Here's how it works:

• We rewrote the original expression \( \log_{5} \frac{3}{2} \) using the division property. This gave us \( \log_{5} 3 - \log_{5} 2 \).
• With the values \( \log_{5} 3 \approx 0.6826 \) and \( \log_{5} 2 \approx 0.4307 \), we substituted and performed the subtraction: \( 0.6826 - 0.4307 \).
• The result is \( 0.2519 \), which gives an approximate value of the original logarithmic expression.

By following these logical steps, we ensure accuracy and clarity in our calculations, leading us to reliable results. This systematic approach transforms potentially daunting logarithmic problems into manageable tasks.