Problem 15 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The approximate value is -0.2519.

Step 1: Understand the Logarithm Quotient Rule

The logarithm quotient rule states that for any positive numbers \(a\) and \(b\), and any base \(c\), \(\log_c \left(\frac{a}{b}\right) = \log_c a - \log_c b\). This formula allows you to break down the logarithm of a quotient into the difference of two logarithms.

Step 2: Apply the Quotient Rule

Apply the logarithm quotient rule to the expression \(\log_5 \left(\frac{2}{3}\right)\). It becomes: \(\log_5 2 - \log_5 3\).

Step 3: Substitute Known Values

Using the given values, substitute \(\log_5 2 \approx 0.4307\) and \(\log_5 3 \approx 0.6826\) into the equation. This results in: \(0.4307 - 0.6826\).

Step 4: Calculate the Difference

Subtract the two logarithm values: \(0.4307 - 0.6826 = -0.2519\). This is the approximate value of \(\log_5 \left(\frac{2}{3}\right)\).

Key Concepts

Logarithm Quotient Rule

The logarithm quotient rule is a useful tool for simplifying the logarithm of a fraction. If you have two numbers, \(a\) and \(b\), and you want to find the logarithm of their quotient, you can use this rule to break it down into easier parts. The rule is expressed as: \[ \log_c \left(\frac{a}{b}\right) = \log_c a - \log_c b \] This means you can take the logarithm of each number individually and then subtract one from the other. This is an especially powerful technique when dealing with logarithms where direct calculation is complex or not possible.Using this rule, complex logarithmic calculations become simpler, as seen in our example problem. It's important to remember that this rule requires both \(a\) and \(b\) to be positive and the same base \(c\) for the calculation to be valid.

Logarithmic Expressions

Logarithmic expressions are mathematical expressions that involve logarithms. They can be in the form of single logs or combinations of several, such as sums, differences, or products of logs. Understanding how to manipulate these expressions is fundamental to solving many problems in mathematics. Key properties of logarithms that allow us to handle these expressions include:

• The product rule: \(\log_c (a \cdot b) = \log_c a + \log_c b\)
• The power rule: \(\log_c (a^b) = b \cdot \log_c a\)
• The change of base formula: \(\log_b a = \frac{\log_c a}{\log_c b}\)

These properties equip us to re-write logarithmic expressions into forms that are easier to manage and solve. In the original exercise, the focus was on using the quotient rule to simplify \(\log_5 \left(\frac{2}{3}\right)\), breaking it into separate logarithms of its numerator and denominator.

Approximation of Logarithms

Approximating logarithms is a practical skill, especially when dealing with logs of numbers that are not whole or simple roots. In many cases, we use given approximations to calculate the values needed. Consider the provided values in the problem: \(\log_5 2 \approx 0.4307\) and \(\log_5 3 \approx 0.6826\). These are not exact values but close enough approximations that allow us to perform further calculations.When you have accurate approximations, you can perform operations like addition or subtraction to find other related logarithmic values, as shown in the exercise. It’s essential to ensure that these approximations are as precise as possible for the best results. Approximations find widespread use, such as in calculators and computer algorithms, where exact computation might be impractical or unnecessary for the user's needs.