Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{10} \frac{y}{2}\)

Step-by-Step Solution

Answer

\(\log_{10}y - \log_{10}2\)

Step 1: Apply the logarithm quotient rule

The log of a quotient can be written as a difference of the referenced logs. Using the mentioned property, \(\log _{10} \frac{y}{2}\) can be rewritten as \(\log_{10}y - \log_{10}2\).

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Key Concepts

Logarithm Quotient Rule

Understanding the logarithm quotient rule is essential when working with logarithmic expressions involving division. This rule states that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. In mathematical terms, for any positive real numbers a, b, and base c, where c is not equal to 1, we can express this as:
\[\begin{equation}\log_c\left(\frac{a}{b}\right) = \log_c(a) - \log_c(b)\end{equation}\]
Let's apply this rule to an example: If given the expression \(\log_{10} \frac{y}{2}\), we recognize that y is the numerator and 2 is the denominator. According to the quotient rule, this can be simplified to \(\log_{10}y - \log_{10}2\). This rule is particularly powerful because it breaks down a single logarithmic expression into a difference of two simpler logarithmic terms, making it easier to work with and understand.

Expanding Logarithmic Expressions

Expanding logarithmic expressions is a technique used to rewrite complex logs into a combination of simpler ones. This often involves applying a series of logarithm rules, not just the quotient rule.

• Product Rule: States that the log of a product is equal to the sum of the logs of the factors; \(\log_c(ab) = \log_c(a) + \log_c(b).\)
• Power Rule: Shows that the log of a power can be expressed as the exponent times the log of the base; \(\log_c(a^k) = k \cdot \log_c(a).\)

Employing these rules can significantly simplify the process of solving equations involving logarithms. In particular, the power rule often complements the quotient rule when dealing with logarithmic expressions that involve powers. This is why understanding each property is vital: they serve as building blocks that, when combined, allow for the expanding and simplifying of any logarithmic expression.

Logarithm Rules

In addition to the quotient rule, there are other fundamental logarithm rules that are crucial when manipulating logarithmic equations. These rules form the backbone of understanding logarithms and their properties. Here are the basics summarized:

• Product Rule: As mentioned, this expresses the log of a product as the sum of the logs (\(\log_b(mn) = \log_b(m) + \log_b(n)\)).
• Quotient Rule: We've already seen how this can express the log of a quotient as the difference of logs.
• Power Rule: This allows rewriting the log of an exponent as the exponent multiplied by the log of the base (\(\log_b(m^n) = n\cdot\log_b(m)\)).

In all cases, it's assumed that the bases are the same and are neither 0 nor 1. By mastering these rules, one can expand, condense, and transform logarithmic expressions to proceed with solving more advanced mathematics problems. It’s important when using these rules to always ensure the variables are within the domain where the logarithms are defined, typically this means making sure the log arguments are positive numbers.