Problem 90 - College Algebra and Calculus: An Applied Approach

Question

Condense the expression to the logarithm of a single quantity.\(\log _{10} 4-\log _{10} z\)

Step-by-Step Solution

Answer

\(\log_{10} (4 / z)\

Step 1: Identify the laws of logarithms

Understand the properties of logarithms, especially that the difference between two of the same base can be represented as the quotient of their arguments, commonly stated as \(\log_b a - \log_b c = \log_b (a/c)\).

Step 2: Apply the law of logarithms

In the expression \( \log_{10} 4 - \log_{10} z \), since both have the same base (which is 10), apply the above-mentioned property. The expression can be written as \(\log_{10} (4 / z)\).

Step 3: Write down the final answer

The final representation of the expression \( \log_{10} 4 - \log_{10} z \) as the log of a single number is \(\log_{10} (4 / z)\).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make them useful for simplifying expressions and solving equations. Key properties include:

• The product property: Which states that the logarithm of a product is equal to the sum of the logarithms of the factors. That is, \( \log_b (xy) = \log_b x + \log_b y \).
• The quotient property: Which allows us to state that the logarithm of a quotient is the difference of the logarithms. For example, \( \log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c \).
• The power property: This property helps us when dealing with exponents, stating \( \log_b (x^y) = y \log_b x \).

Understanding these properties aids in the manipulation and simplification of logarithmic expressions, such as turning a subtraction of logs into a single logarithm, as shown with the expression \( \log_{10} 4 - \log_{10} z \).

Logarithm Rules

Logarithm rules are crucial for simplifying complex expressions and solving logarithmic equations. They revolve around the properties mentioned earlier but apply more broadly in mathematical computations. The basic rules you often use include:

• Change of base rule: Used to evaluate logarithms in different bases, \( \log_b a = \frac{\log_k a}{\log_k b} \) for a new base \( k \).
• Identity rule: Stating that \( \log_b b = 1 \) since \( b^1 = b \).
• Zero property: Relating that \( \log_b 1 = 0 \) because any non-zero number raised to the power zero equals 1.

These rules are like tools in your mathematical toolbox, enabling you to transform and solve logarithmic equations more efficiently. By applying these rules, expressions such as \( \log_{10} 4 - \log_{10} z \) become much simpler to handle using the quotient property.

Logarithm Base 10

Logarithm base 10, often called the common logarithm, is widely used because of its simplicity and frequent appearance in real-world applications. It simplifies calculations involving powers of ten, which align with the decimal system that is universally used in measurements and scientific notations.

• Properties: Functions similarly to other bases but offers practical convenience with the common use cases in science and engineering.
• Simplification: Commonly simplifies statements involving large and small quantities by expressing them as powers of 10.
• Calculator Compatibility: Most scientific calculators are built with 'log' button initialized to base 10 by default.

In mathematics education, base 10 logarithms provide a basis for understanding more complex bases and exponential growth patterns. With these, an expression like \( \log_{10} (4 / z) \) represents part of a broader range of practical applications.