Problem 30 - Algebra 2

Question

Expand each logarithm. \(\log _{b} \frac{1}{x}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithm \( \log_b (1/x) \) is \( - \log_b x \).

Step 1: Identify the property of logarithms to apply

The logarithm given in the question is of the form \( \log_b (1/x) \), for which we will use the property that \( \log_b (1/x) = - \log_b x \). This property allows us to handle the fraction inside the logarithm.

Step 2: Apply the property of logarithms

Using the identified property, the given logarithm \( \log_b (1/x) \) can be rewritten as \( - \log_b x \).

Key Concepts

Logarithm Properties

Logarithms have several useful properties that make expanding and simplifying expressions much easier. These properties help us handle complex expressions involving logarithms:

• Product Property: The logarithm of a product is the sum of the logarithms of the factors. Mathematically, this is written as \(\log_b (xy) = \log_b x + \log_b y\).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms of the numerator and denominator. This can be expressed as \(\log_b (x/y) = \log_b x - \log_b y\).
• Power Property: The logarithm of a number raised to a power is the power times the logarithm of the number. This is given by \(\log_b (x^c) = c \log_b x\).
• Reciprocal Property: This property states that \(\log_b (1/x) = -\log_b x\). It simplifies the expression by changing the logarithm of a reciprocal to a negative logarithm.

The reciprocal property, used in the original solution, is particularly helpful when dealing with fractions inside a logarithm. By recognizing and applying these properties, logarithmic expressions become less intimidating to work with.

Expansion of Logarithms

The process of expanding logarithms involves rewriting a logarithmic expression into a more accessible or simplified form using the properties of logarithms. This is particularly useful in equations or transformations where we need to isolate terms or simplify expressions for further calculation.
To expand a logarithm effectively:

• Identify any product, quotient, or power in the logarithmic expression that can be simplified using logarithm properties.
• Apply the appropriate logarithmic property to break down the expression into simpler parts. For example, use the quotient property \(\log_b (x/y) = \log_b x - \log_b y\) to handle division inside the logarithm.
• Continue applying properties until the expression is fully simplified. Each step reduces the complexity of the logarithm and brings clarity to the structure of the expression.

In the given problem, \(\log_b (1/x)\) was expanded to \(-\log_b x\) using the reciprocal property. This form is often more useful for analytical manipulation in problems or studies involving logarithms.

Exponent Rules

Exponential and logarithmic functions are deeply interconnected, wherein exponent rules often complement logarithmic properties. Understanding these rules provides the foundation for manipulating expressions in both exponential and logarithmic forms.

• Product of Powers Rule: When multiplying like bases, add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing like bases, subtract their exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Power of a Product Rule: When raising a product to an exponent, apply the exponent to each factor: \((ab)^n = a^n b^n\).
• Common Logarithmic Relationship: Since logarithms are inverses of exponentials, knowing that \(b^{\log_b x} = x\) and \(\log_b (b^x) = x\) can be immensely helpful in simplification steps.

In logarithmic problems like the one above, understanding exponent rules assists in comprehending how logarithmic properties transform expressions. Oftentimes, the simplification of logarithms is hand-in-hand with converting between exponential and logarithmic forms.