Problem 84 - College Algebra Essentials

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). \(\log _{b} 6\)

Step-by-Step Solution

Answer

\(\log _{b} 6 = A + C\).

Step 1: Breakdown of \(\log _{b} 6\)

\(\log _{b} 6\) can be expressed as the sum of \(\log _{b} 2\) and \(\log _{b} 3\), by applying the product rule of logarithms. So, \(\log _{b} 6 = \log _{b} 2 + \log _{b} 3\).

Step 2: Substitution

Substitute given values \(A = \log _{b} 2\) and \(C = \log _{b} 3\) into the equation from step 1. So, \(\log _{b} 6 = A + C\).

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a fundamental concept in logarithmic operations. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. In mathematical terms, for any positive numbers \( x \), \( y \), and a base \( b \), this can be expressed as:

\[ \log_b(xy) = \log_b(x) + \log_b(y) \]

This property allows us to simplify expressions that involve the product of numbers by breaking them down into smaller, more manageable pieces.

• Example: If you have \( \log_b(6) \), you can write 6 as the product of 2 and 3.
• Then apply the product rule: \( \log_b(6) = \log_b(2 \cdot 3) = \log_b(2) + \log_b(3) \).

By using the product rule, calculations become simpler, and it's easier to work with and understand complex logarithmic expressions. This rule is particularly useful for algebraic manipulations involving logarithms, as it allows us to translate products into sums, which are usually easier to compute or estimate.

Logarithmic Expressions

Logarithmic expressions are mathematical expressions that involve the use of logarithms. They are often used to find the exponent to which a base number must be raised to obtain a certain number. In its simplest form, a logarithmic expression looks like \( \log_b(x) \), where \( b \) is the base and \( x \) is the number.

• Components: A logarithm consists of a base \( b \), an argument \( x \), and the result is the power to which the base must be raised to produce the argument.
• Notation: \( \log_b(x) \) means the logarithm of \( x \) with base \( b \).

Logarithmic expressions can be transformed using various logarithmic rules such as the product rule, quotient rule, and power rule. Let's consider how these expressions can be handled:

• They can be simplified to a form that is easier to work with.
• They enable you to solve for unknowns when they appear in equations, through algebraic manipulation.

Mastering the ability to manipulate and understand logarithmic expressions enhances your algebraic skills and prepares you for more advanced topics such as exponential equations and calculus.

Algebraic Manipulation

Algebraic manipulation involves rewriting expressions in a different form using algebraic rules and properties. When working with logarithms, algebraic manipulation can transform difficult problems into simpler ones by breaking down expressions, substituting known values, or applying logarithmic properties.

• Simplification: Use known values or substitutions to simplify complex expressions.
• Rearrangement: Apply properties like the product, quotient, or power rule to rearrange terms.
• Substitution: Replace variables with known values to evaluate the expressions easily.

In the provided exercise, algebraic manipulation helped by substituting the values \( A = \log_b(2) \) and \( C = \log_b(3) \) into \( \log_b(6) \), making the complex expression simple and understandable:- Originally, \( \log_b(6) \) was transformed by recognizing it as the sum \( \log_b(2) + \log_b(3) \), thanks to the product rule of logarithms.- Then, known values were substituted: \( \log_b(6) = A + C \).These techniques are powerful tools for tackling advanced mathematical problems, providing a pathway from complex, abstract ideas to concrete, clear solutions.