Problem 39 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58\) , 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

\(4 \cdot \log _{8} x\)

Step 1: Understand the Power Rule of Logarithms

When a logarithm is raised to a power, this can be rewritten by taking that power and multiplying it with the logarithm. In other words, for any positive number 'a', 'x', and 'n' not equal to 1, the rule is expressed as \(\log _{a} x^{n} = n \cdot \log _{a} x\).

Step 2: Apply the Power Rule

Applying this rule to the given expression \(\log _{8} x^{4}\), the power 4 can be taken out and multiplied with the logarithm to expand the expression. This gives \(4 \cdot \log _{8} x\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when working with logarithmic expressions. These properties are derived from the definitions of logarithms and the rules of exponents.

There are three commonly used properties that can help simplify complex logarithmic expressions. First, the product rule, which states that \(\log_a(xy) = \log_a(x) + \log_a(y)\). This property implies that the logarithm of a product can be expressed as the sum of logarithms. Next, the quotient rule, which is given by \(\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\). As the product rule’s counterpart, it allows the logarithm of a division to be represented as a difference. Lastly, the power rule, which we'll explore in more detail in the next section. The power rule is especially helpful in expanding logarithmic expressions where a variable is raised to a power.

• Product Rule: \(\log_a(xy) = \log_a(x) + \log_a(y)\)
• Quotient Rule: \(\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\)
• Power Rule: \(\log_a(x^n) = n\cdot\log_a(x)\)

By applying these properties, you can transform complex logarithmic expressions into simpler forms that can be more easily evaluated or compared.

Power Rule of Logarithms

The power rule of logarithms is a transformative tool that can be used to manipulate and expand logarithmic expressions involving exponents.

This rule states that for any positive base 'a', variable 'x', and exponent 'n', the logarithm of 'x' raised to the power of 'n' is the same as multiplying 'n' by the logarithm of 'x'. The formal expression of the power rule is \(\log_a(x^n) = n\cdot\log_a(x)\). Here’s a step-by-step breakdown:

1. Identify the exponent 'n' in the logarithmic expression.
2. Move the exponent from its position on 'x' to the front of the logarithm, converting it into a multiplicative factor.

For instance, in the exercise \(\log_8(x^4)\), the power rule allows us to take the exponent 4 and multiply it by the logarithm, simplifying the expression to \4\cdot\log_8(x)\.

This property not only makes expressions less complex but also prepares them for further simplification or computation, particularly when dealing with logarithmic equations or inequalities.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and have wide applications in various scientific fields, including physics, chemistry, and information theory. These functions model scenarios of growth and decay and help in determining the exponents necessary to reach a certain value in an exponential equation.

The general form of a logarithmic function is \(y = \log_a(x)\), where 'a' is the base, 'x' is the argument, and 'y' is the result or logarithm. The domain of this function is \(x > 0\) because the logarithm of a negative number or zero is undefined in real numbers.

Characteristics of Logarithmic Functions:

• Their graphs have a vertical asymptote at \(x = 0\).
• They pass through the point (1,0) regardless of the base 'a' as \(\log_a(1) = 0\).
• They increase as 'x' increases if the base 'a' is greater than 1.
• They decrease as 'x' increases if the base 'a' is between 0 and 1.

Understanding the nature and behavior of logarithmic functions is essential when working with logarithms. They allow us to rewrite exponential relations in a form that is often more convenient for analysis and solution, particularly when solving problems involving growth rates and half-lives in science or calculating the time complexity in algorithms.