Problem 7 - College Algebra

Question

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. $$ \ln \left(\frac{1}{4^{\mathrm{k}}}\right) $$

Step-by-Step Solution

Answer

The expanded form is \(-k \cdot \ln(4)\).

Step 1: Apply Logarithm Power Rule

First, identify the power rule of logarithms for the expression \( \ln \left(\frac{1}{4^{k}}\right) \). According to the power rule, \( \ln(a^b) = b \cdot \ln(a) \). In this case, apply this to the denominator, so you have \( \ln \left(4^{-k}\right) \).

Step 2: Apply Negative Exponent Rule for Logarithms

Recognize that \( 4^{-k} \) is equivalent to \( \frac{1}{4^k} \). Applying the negative exponent rule, \( \ln(4^{-k}) = -k \cdot \ln(4) \).

Step 3: Simplify the Expression

Replace the logarithm of the division with a product of a logarithm and a scalar, yielding \( -k \cdot \ln(4) \). This is already expanded in terms of a product of a scalar and a logarithm.

Key Concepts

Power Rule in Logarithms

The power rule in logarithms is one of the fundamental properties that simplifies expressions involving exponents. When we come across the power rule, it is expressed as \( \ln(a^b) = b \cdot \ln(a) \). This rule tells us that when a number inside a logarithm has an exponent, we can "move" the exponent in front of the logarithm as a multiplier. This is incredibly useful for simplifying expressions that involve powers.Let's consider a practical example: if we have \( \ln(9^2) \), by applying the power rule, this can be rewritten as \( 2 \cdot \ln(9) \). By transforming the expression with this rule, it becomes much easier to handle, especially when you need to perform further calculations or expansions.Using the power rule helps in breaking down complex logarithm expressions into more manageable terms, letting you find solutions efficiently and accurately.

Negative Exponent Rule

The negative exponent rule is a vital concept in both algebra and logarithms. With this rule, any number with a negative exponent indicates that the number is actually a fraction. Specifically, \( a^{-b} = \frac{1}{a^b} \).This has essential implications for logarithmic expressions. For instance, in the expression \( \ln(4^{-k}) \), by using the negative exponent rule, it is simplified to \( \ln\left(\frac{1}{4^k}\right) \). This makes it clearer that the expression \( 4^{-k} \) in its original form is already an operation of division, where the base is in the denominator.Applying the negative exponent rule helps in rewriting expressions to make patterns more visible, which can be crucial when working with complex equations. It's also a steppingstone to integrating additional logarithmic properties seamlessly.

Logarithmic Identities

Logarithmic identities are the backbone of working with logarithms. They provide a set of rules that allow us to manipulate and transform logarithmic expressions efficiently. These identities include the power rule, the negative exponent rule, and more.One important identity is the product rule, which states that \( \ln(a \cdot b) = \ln(a) + \ln(b) \). Another identity is the quotient rule, expressed as \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \). These identities enable the transformation of multiplications and divisions within logarithms to additions and subtractions among logarithms. This proves incredibly useful when trying to expand or simplify logarithmic expressions.For example, if you need to expand \( \ln\left(\frac{x}{y}\right) \), using the quotient rule gives you \( \ln(x) - \ln(y) \). Logarithmic identities help simplify complex problems into simpler ones, enabling easier computation and a better understanding of the problem at hand. Understanding these identities is key to mastering logarithms in general mathematics and applied contexts.