Problem 255 - Precalculus

Question

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

Step-by-Step Solution

Answer

\( -k \cdot \ln(4) \)

Step 1: Apply the Quotient Rule for Logarithms

The given expression is \( \ln \left( \frac{1}{4^{k}} \right) \). According to the quotient rule of logarithms, \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \). In this case, let \( a = 1 \) and \( b = 4^{k} \). So the expression becomes:\[ \ln(1) - \ln(4^{k}) \].Since \( \ln(1) = 0 \), this simplifies to:\[ 0 - \ln(4^{k}) \].

Step 2: Simplify the Result

From Step 1, we have \( 0 - \ln(4^{k}) \), which simplifies to:\[ -\ln(4^{k}) \].

Step 3: Apply the Power Rule for Logarithms

Using the power rule of logarithms, \( \ln(a^{b}) = b \cdot \ln(a) \). We have \( -\ln(4^{k}) \). Apply the rule where \( a = 4 \) and \( b = k \), so:\[ -k \cdot \ln(4) \].

Key Concepts

Quotient Rule for Logarithms

The quotient rule for logarithms is a fundamental property that allows us to simplify expressions involving the logarithm of a quotient. If you have a logarithm of the form \( \ln \left( \frac{a}{b} \right) \), the quotient rule states that it can be rewritten as \( \ln(a) - \ln(b) \). This is effectively breaking down the complex logarithmic expression into a simpler difference of two logarithms.

• Make sure both \( a \) and \( b \) are positive because the logarithm is only defined for positive numbers.
• The rule is particularly useful for breaking down expressions into manageable parts that can be dealt with separately.

In the original exercise, this rule was used to transform \( \ln \left( \frac{1}{4^k} \right) \) into \( \ln(1) - \ln(4^k) \). Since we know \( \ln(1) \) is always zero, the expression simplifies further. This technique is often the first step in breaking down complex logarithmic expressions.

Power Rule for Logarithms

The power rule for logarithms is another crucial tool for expanding and simplifying logarithmic expressions. It states that any logarithm of a power can be rewritten by bringing the exponent in front as a multiplier: \( \ln(a^b) = b \cdot \ln(a) \). This transformation allows you to handle the exponent more directly by turning it into a coefficient.

• Use this rule only when the base \( a \) is greater than zero and \( a eq 1 \).
• This conversion turns a potentially problematic exponent into a simple multiplication, easing further calculations or transformations.

In the exercise, once the expression was reduced to \( -\ln(4^k) \), the power rule allowed the exponent \( k \) to move in front, resulting in \( -k \cdot \ln(4) \). This simplifies calculations by reducing the problem to basic multiplication.

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions is part of efficiently solving logarithmic problems, and it often involves using the properties and rules of logarithms such as the quotient rule and the power rule. The goal of simplification is to reduce the expression to its simplest form, making it easier to understand or to solve related equations.

• Simplification often involves rewriting expressions to reveal their most fundamental components, typically as sums, differences, or products of individual logarithms.
• This can often lead to further simplification mathematically, or aid in spotting patterns or redundancies in equations.

In the given exercise, simplification involved applying the quotient and power rules sequentially to render the complex logarithmic expression \( \ln \left( \frac{1}{4^{k}} \right) \) into a simple linear form \( -k \cdot \ln(4) \). Each step led to a more manageable expression, making the original problem easier to grasp and solve.