Problem 32 - A Graphical Approach to College Algebra

Question

Simplify each expression. (a) \(\log _{3} 3^{19}\) (b) \(\log _{4} 4^{17}\) (c) \(\log _{12} 12^{1 / 3}\) (d) \(\log _{a} \sqrt{a}(a>0, a \neq 1)\)

Step-by-Step Solution

Answer

(a) 19, (b) 17, (c) 1/3, (d) 1/2

Step 1: Identify the Logarithm Property

Recognize that the expressions given involve logarithms of the form \( \log_b(b^x) \). This is simplified using the property: \( \log_b(b^x) = x \).

Step 2: Simplifying Part (a)

The expression is \( \log _{3} 3^{19} \). Applying the property from Step 1, we have:\[\log _{3} 3^{19} = 19\]

Step 3: Simplifying Part (b)

The expression is \( \log _{4} 4^{17} \). Using the same property, we find:\[\log _{4} 4^{17} = 17\]

Step 4: Simplifying Part (c)

The expression is \( \log _{12} 12^{1 / 3} \). Applying the property gives:\[\log _{12} 12^{1 / 3} = \frac{1}{3}\]

Step 5: Simplifying Part (d)

The expression is \( \log _{a} \sqrt{a} \). Note that \( \sqrt{a} = a^{1/2} \). Thus, applying the property:\[\log _{a} a^{1/2} = \frac{1}{2}\]

Key Concepts

Understanding Logarithm Properties

Logarithm properties are essential tools in simplifying complex mathematical expressions. One of the most important properties is the identity \( \log_b(b^x) = x \). This means that if you take a logarithm with a base that matches the base of the exponent, the result is simply the exponent itself.
For example, in \( \log_3(3^{19}) \), the base of the logarithm matches the base of the exponent, which simplifies directly to 19.

• This property helps turn a potentially confusing calculation into a straightforward one.
• It's particularly useful when dealing with exponential expressions inside logarithms.

Grasping Exponentiation

Exponentiation involves raising a number, called the base, to the power of an exponent. This is a way to denote repeated multiplication of a number by itself.
In the expressions given, such as \( 3^{19} \), the number 3 is multiplied by itself 19 times.

• This concept ties closely with logarithms, as logarithms are essentially the inverse operation to exponentiation.
• Understanding this relationship is key to simplifying logarithmic expressions.

Grasping the inverse nature of logarithms and exponentiation allows you to switch between the two forms effortlessly, making problem-solving much more intuitive.

Simplification Strategies

Simplification in mathematics means reducing an expression to its simplest form. The aim is to make the expression easier to understand or solve.
When simplifying logarithmic expressions, applying known properties like \( \log_b(b^x) = x \) can help streamline the process.
For example, in \( \log_4(4^{17}) \), using the property directly reduces the complexity by simplifying the expression to 17.

• Recognizing patterns and applying the correct property is crucial for effective simplification.
• Simplifying math problems can prevent errors and improve efficiency.

College Algebra and Its Applications

College Algebra often includes topics such as logarithms and exponentiation, which are foundational in more advanced mathematics. Understanding these concepts is essential not only in tests but in practical applications as well.
Whether you are learning logarithmic functions, solving exponential equations, or optimizing computations in coding and algorithms, these skills come into play.

• Mastery of College Algebra provides critical thinking skills applicable to various fields.
• It offers the ability to solve real-world problems using mathematical reasoning.

Achieving proficiency in these areas can greatly enhance your confidence and capability in tackling mathematical challenges.