Problem 31 - A Graphical Approach to College Algebra

Question

Simplify each expression. (a) \(3^{\log _{3} 7}\) (b) \(4^{\log _{4} 9}\) (c) \(12^{\log _{13} 4}\) (d) \(a^{\log _{c} k}(k > 0, a > 0, a \neq 1)\)

Step-by-Step Solution

Answer

(a) 7; (b) 9; (c) No simplification; (d) No simplification.

Step 1: Understand the Problem

In each exercise, we are given an expression of the form \(a^{\log_{a} b}\) or \(a^{\log_{c} k}\). Our task is to simplify these expressions.

Step 2: Apply the Exponent-Logarithm Identity for Same Base

The identity \(a^{\log_{a} b} = b\) can be applied whenever the base of the exponent and the base of the logarithm are the same. This simplifies the expression directly to the argument of the logarithm, \(b\).

Step 3: Simplify Part (a)

Given expression \(3^{\log_{3} 7}\), we apply the identity from Step 2. Since the base of the exponent (3) matches the base of the logarithm (also 3), the expression simplifies to \(7\).

Step 4: Simplify Part (b)

Given expression \(4^{\log_{4} 9}\), we apply the identity. The base of the exponent (4) matches the base of the logarithm (also 4), so the expression simplifies to \(9\).

Step 5: Consider Part (c)

Given expression \(12^{\log_{13} 4}\), we notice the bases of the exponent (12) and the logarithm (13) are different. This prevents direct application of the identity used in earlier steps. In many cases, this expression is left as it is or interpreted based on the logarithm properties but can't be simplified directly without additional context.

Step 6: Simplify Part (d) Using Variables

For the expression \(a^{\log_{c} k}\), since \(a\) and \(c\) can be any positive number, we cannot directly apply the identity. Similar to part (c), this expression is not simplified without specific values or context determining a relationship between \(a\) and \(c\).

Key Concepts

Simplifying Expressions

When simplifying expressions, particularly those involving exponent and logarithm identities, the goal is to reduce complexity while retaining mathematical equivalence.
Here, we work with expressions involving exponents, where the base and the logarithm share the same number. Applying the exponent-logarithm identity, \(a^{\log_{a} b} = b\), allows us to directly simplify expressions to the argument of the logarithm when the bases are the same. This identity is powerful because it demonstrates the inverse nature of exponential and logarithmic functions.

• Example (a): Given \(3^{\log_{3} 7}\), we notice the base of both exponent and log is 3, so the expression simplifies directly to \(7\).
• Example (b): For \(4^{\log_{4} 9}\), the matching bases 4 allow the simplification to \(9\).

However, when the bases differ, as in parts (c) and (d), simplification through this identity isn't straightforward. This is an essential aspect when dealing with algebraic expressions, highlighting the importance of understanding the conditions required for certain simplification steps.

Logarithmic Equations

Logarithmic equations involve the use of logarithms to determine unknown quantities and often require rewriting terms to solve them.
A key strategy for simplifying certain types of logarithmic equations is recognizing when you can apply the identity \(a^{\log_{a} b} = b\), especially within mathematical expressions.

• In the context of the given exercise, equations like \(3^{\log_{3} 7} = 7\) show how logarithms, when paired with a matching base exponent, essentially 'cancel' each other out.
• For equations where bases differ, like \(12^{\log_{13} 4}\), further manipulation or interpretation using properties such as change of base might be required, although direct simplification using the identity isn't possible.

Recognizing these equations allows us to manipulate and solve logarithm problems efficiently by aligning our approach to the given equation's structure. Exploring different logarithmic properties can be crucial, especially in situations where simple identities don't apply directly, offering insight into more complex algebraic relationships.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a value. Simplifying these expressions requires understanding how to manipulate their components while maintaining balance and equivalence.
In the exercise, simplifying expressions like \(a^{\log_{c} k}\) involves recognizing when simplification is straightforward and when it requires deeper insights or assumptions.

• Expressions like \(a^{\log_{c} k}\) highlight the importance of the bases. Without specific values or context, these expressions often cannot be simplified via direct application of logarithmic identities.
• Algebraic expressions with logarithms become simpler when we understand base and argument relationships, emphasizing the need to consider each component's role, such as variables signaling undefined parameters unless further specified.

Mastering algebraic manipulations, especially with logarithms, empowers problem solving by allowing us to find solutions using a structure of known identities and properties. This clarity in algebraic thinking is central to both mathematics and applied scenarios.