Problem 85 - College Algebra Essentials

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). \(\log _{b} 8\)

Step-by-Step Solution

Answer

The expression \(\log _{b} 8\) in terms of \(A\) and \(C\) is \(3A\).

Step 1: Express 8 as a power of 2

Express the number 8 as a power of 2 to get \(\log _{b} (2^3)\).

Step 2: Use the properties of logarithms

Substitute \(\log _{b} (2^3)\) using the properties of logarithms to get \(3 \log _{b} 2\).

Step 3: Replace \(\log _{b} 2\) with \(A\)

Finally, replace \(\log _{b} 2\) with the given \(A\) to find the logarithm in terms of \(A\), to get \(3A\).

Key Concepts

Understanding Logarithmic Expressions

Logarithms can seem intimidating at first glance, but understanding them is crucial for solving many problems in mathematics. A logarithmic expression, such as \( \text{log}_{b} x \), can be interpreted as the power to which you need to raise the base \( b \) to get \( x \). In the given exercise,
\( \text{log}_{b} 2 = A \) and \( \text{log}_{b} 3 = C \), this concept is applied by breaking down complex logarithmic expressions into simpler terms using the known values \(A\) and \(C\).

Logarithmic expressions follow specific rules or properties that allow us to manipulate them with ease. One such property is the power rule, which states that \( \text{log}_{b} (x^k) = k \text{log}_{b} x \). This property was used in the exercise to rewrite \( \text{log}_{b} 8 \) as \( \text{log}_{b} (2^3) \) and subsequently as \( 3 \text{log}_{b} 2 \) by expressing 8 as a power of 2, which is a fundamental skill when dealing with logarithmic expressions.

Exponentiation and its Role in Logarithms

Exponentiation is a form of repeated multiplication and plays a key role in the realm of logarithms. It is the operation of raising a number (the base) to a certain power (the exponent), which represents how many times the base is multiplied by itself.
For instance, when we describe \(8\) as \(2^3\), we indicate that \(2\) is multiplied by itself three times. This understanding is necessary when we encounter a logarithm of a power, like \(\log _{b} 8\) in our exercise.

By utilizing the knowledge that \(8\) is \(2\) raised to the third power, we can simplify the expression using the rule of exponentiation within logarithms. The simplification used the concept that a logarithm of a number raised to an exponent is equal to that exponent times the logarithm of the base number. This forms the basis for simplifying complex logarithmic expressions and solving them in terms of known variables.

Algebraic Manipulation in Logarithmic Context

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This skill is indispensable when working with logarithmic expressions, enabling you to express one term in terms of others or to isolate a variable.
In the exercise, once we understood that \(\log _{b} 8\) can be written as \(\log _{b} (2^3)\), we made use of algebraic manipulation by applying logarithmic properties to simplify the expression. The expression \(3 \log _{b} 2\) was then further simplified by substituting \(\log _{b} 2\) with the given variable \(A\), resulting in \(3A\).

By practicing algebraic manipulation in logarithmic contexts, such as combining like terms or using substitution as showcased in the solution, students can effectively manage even the most complex of logarithmic expressions. The key is to remember the fundamental properties of logarithms, associate them with basic algebraic principles, and apply this knowledge to transform the expressions step by step.