Problem 15 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b} x^{3} $$

Step-by-Step Solution

Answer

The expanded form of \(\log_{b} x^{3}\) is \(3 \cdot \log_{b}x\).

Step 1: Apply the power rule

The given expression is \(\log_{b} x^{3}\). Here, \(x\) is raised to the power of 3. According to the power rule of logarithms, we can move this power in front of the logarithmic expression. So, the expanded form becomes \(3 \cdot \log_{b}x\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The expanded form of \(\log_{b} x^{3}\) is \(3 \cdot \log_{b}x\).

Key Concepts

Logarithmic Expressions

To understand how to manipulate logarithms, it's crucial to start with a clear grasp of what logarithmic expressions entail. A logarithmic expression, like \( \log_b x \), is the inverse operation of raising a base \( b \) to a power to get \( x \). Essentially, it answers the question: 'To what power must we raise \( b \) to obtain \( x \)?' Understanding this inverse relationship between exponents and logarithms can make it easier to work with logarithmic functions and solve logarithmic equations.

Logarithmic functions are used throughout various disciplines, from calculating the magnitude of earthquakes (Richter scale) to measuring the acidity or alkalinity of solutions (pH scale). Hence, developing strong skills in manipulating logarithmic expressions can have broad applications beyond the classroom.

Power Rule of Logarithms

The power rule for logarithms is a valuable property that allows for the simplification of logarithmic expressions involving exponents. It states that a logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number itself. Mathematically, it can be expressed as \( \log_b (x^p) = p \cdot \log_b x \).

As demonstrated in the original exercise, applying the power rule to \( \log_{b} x^{3} \) results in the expression \( 3 \cdot \log_{b}x \). This rule is particularly useful when you need to expand or simplify logarithmic expressions, and it can also be used in reverse to combine logarithmic terms into a single term.

Expanding Logarithms

Expanding logarithms refers to the process of taking a complex logarithmic expression and breaking it down into simpler parts using various logarithmic properties, such as the power rule, product rule, and quotient rule. The goal is to transform the logarithm of a product, quotient, or power into a sum or difference of logarithms, which can often be easier to evaluate or integrate into larger problem-solving steps.

For example, along with the power rule, you might encounter the product rule \( \log_b(xy) = \log_b x + \log_b y \) or the quotient rule \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \). These properties allow for the expansion of more complex logarithmic expressions, making it easier to analyze and understand the underlying relationships between the logged values.

Evaluating Logarithms Without a Calculator

Sometimes, you might need to evaluate logarithmic expressions without the assistance of a calculator, thus it's important to know a few tricks. Firstly, understanding the relationship between logarithms and exponents is key. You should be comfortable with common bases, such as 10 and \( e \), and their related powers. Additionally, keeping in mind some logarithmic identities, such as \( \log_b 1 = 0 \) and \( \log_b b = 1 \) can simplify evaluations substantially.

When facing an expression like \( \log_b x \), remember that if \( x \) is a power of \( b \), you can determine the logarithm's value directly. For instance, if \( b = 10 \) and \( x = 100 \) then \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \). With practice, evaluating logarithms without a calculator becomes a straightforward task, enhancing your mental math skills and understanding of logarithmic relationships.