Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{8} x^{4}$$.

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression is \( 4 \log_{8}{x} \).

Step 1: Apply the power rule of logarithms

The given expression is \( \log_{8}{x^{4}} \). We apply the power rule, which states that \( \log_{b}{a^{n}} = n \log_{b}{a} \). Hence, \( \log_{8}{x^{4}} \) will be converted into \( 4 \log_{8}{x} \).

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 the logarithmic expression is \( 4 \log_{8}{x} \).

Key Concepts

Logarithm Power Rule

Understanding the logarithm power rule is essential for working with logarithmic expressions efficiently. In essence, the power rule tells us that a logarithm of an exponent can be simplified by bringing the exponent in front of the logarithm as a coefficient. For instance, if you encounter an expression like \( \text{log}_b(x^n) \), you can apply the power rule by rewriting it as \( n \times \text{log}_b(x) \).

This rule not only allows for simplification but also makes it easier to handle complex logarithmic equations, particularly during expansion or condensation processes. It's important to remember that to use this rule correctly, the base \( b \) of the logarithm must remain the same throughout the operation, which indicates that the property is being applied to the exponent of the argument \( x \) directly.

Logarithmic Expressions

Logarithmic expressions can seem intimidating at first, but they operate on straightforward principles. A logarithmic expression represents the exponent to which a base must be raised to produce a certain number. It's denoted as \( \text{log}_b(a) \), where \( b \) is the base, and \( a \) is the result of raising \( b \) to the exponent the logarithm is trying to find.

When working with these expressions, several properties of logarithms vastly simplify problem-solving: the power rule, product rule, quotient rule, and change of base formula. Every property has a specific use-case scenario, such as expanding, condensing, or even changing the base of the logarithmic expression for comparison or calculation purposes. Grasping these concepts helps in solving algebraic equations involving logarithms, understanding exponential growth and decay models, and analyzing logarithmic scales in science and engineering fields.

Expanding Logarithms

Expanding logarithms is a crucial skill when solving logarithmic equations or simplifying complex logarithmic expressions. By using the properties of logarithms, you can break down a logarithm into a sum, difference, or product of simpler parts.

For example, to expand the logarithm \( \text{log}(ab) \), you can use the product rule to express this as \( \text{log}(a) + \text{log}(b) \). Similarly, the quotient rule allows you to write \( \text{log}(a/b) \) as \( \text{log}(a) - \text{log}(b) \). This process of expanding is extremely beneficial as it turns multiplicative relationships into additive ones, which are often much more manageable to work with.

Knowing how to properly expand logarithms also lays the groundwork for other mathematical operations, such as solving for unknown variables or integrating complex expressions in calculus.