Problem 12 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}\left(\frac{125}{y}\right) $$

Step-by-Step Solution

Answer

The expanded logarithmic expression is \(3 - \log_{5} y\)

Step 1: Apply the quotient rule

Start by applying the quotient rule which says that the log of a quotient equals the difference of the logs. It turns \(\log _{5}\left(\frac{125}{y}\right)\) into \(\log _{5}(125) - \(\log _{5}(y)\)

Step 2: Convert 125 to base 5

We know that \(5^3 = 125\), so \(\log_{5} 125 = 3\). Hence, the expression now becomes \(3 - \log_{5} y\).

Step 3: Final expression

Therefore, \( \log _{5}\left(\frac{125}{y}\right) = 3 - \log_{5} y\) This is as simplified as this expression can become. It's not possible to evaluate the entire expression without knowing the value of y, as it depends on y.

Key Concepts

Properties of Logarithms

The properties of logarithms are essential tools in algebra to simplify and solve logarithmic expressions. These properties leverage the fundamental nature of logarithms as inverse operations to exponentiation.Some of the primary properties include:

• The Product Rule: This states that the logarithm of a product is the sum of the logarithms, i.e., \( \log_b(mx) = \log_b(m) + \log_b(x) \)
• The Quotient Rule: The logarithm of a quotient is the difference of the logarithms, i.e., \( \log_b\left(\frac{m}{x}\right) = \log_b(m) - \log_b(x) \)
• The Power Rule: This suggests that the logarithm of a power equals the exponent times the logarithm, i.e., \( \log_b(m^x) = x \cdot \log_b(m) \)

These rules are incredibly helpful when expanding or simplifying logarithmic expressions. By understanding and applying these properties, you can manipulate and solve logarithmic equations more efficiently.

Quotient Rule

The quotient rule is one of the core properties of logarithms and a key strategy for simplifying and expanding logarithmic expressions. This rule provides a method for transforming the logarithm of a division into the subtraction of two logarithms. For example, if you have an expression \( \log_b\left(\frac{a}{b}\right) \), you can rewrite it as \( \log_b(a) - \log_b(b) \).In the exercise, the expression \( \log_5\left(\frac{125}{y}\right) \) is simplified using the quotient rule, making it \( \log_5(125) - \log_5(y) \). This transformation is particularly useful when further computations or evaluations need individual terms, such as \( \log_5(125) \). Utilizing the quotient rule is about breaking down complex expressions into more manageable parts.

Logarithm Base Conversion

Converting between bases in logarithms is sometimes necessary to simplify or solve an expression. Knowing the relationship between numbers and their powers helps determine logarithmic values directly. In our example, we simplify \( \log_5(125) \) by recognizing 125 as a power of 5, specifically \( 5^3 \). Thus, \( \log_5(125) \) can be converted to \( 3 \), since logarithms check how many times a base must multiply to reach a particular number. In this case, \( 5^3 = 125 \), making \( \log_5(125) = 3 \). Understanding base conversion can expedite computations without a calculator and deepen comprehension of logarithmic functions.

Expand Logarithmic Expressions

Expanding logarithmic expressions involves breaking them down into the simplest form using the properties of logarithms. The primary aim is to make the expression easier to handle, either manually or conceptually. In the given exercise, expanding \( \log_5\left(\frac{125}{y}\right) \) was achieved by applying the quotient rule and recognizing base powers. From the original expression, \( \log_5(125) - \log_5(y) \) is the expanded version where each component is distinctly identified. This expanded form is not only more straightforward for substitution but also provides clarity whether you evaluate it or perform additional algebraic manipulations. Expanding logarithms helps highlight patterns, simplify complex calculations, and prepares expressions for solving equations.