Problem 7
Question
For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\ln \left(\frac{1}{4^{k}}\right)\)
Step-by-Step Solution
Verified Answer
\( \ln \left( \frac{1}{4^k} \right) = -k \cdot \ln(4) \).
1Step 1: Apply the Power Rule of Logarithms
The power rule states that \( \ln(a^b) = b \cdot \ln(a) \). In this case, we identify \( a = 4 \) and \( b = -k \) since \( \frac{1}{4^k} = 4^{-k} \). Therefore, \( \ln(4^{-k}) = -k \cdot \ln(4) \).
2Step 2: Express as a Product of Logarithms
After applying the power rule and rewriting the expression, we find that the expanded form of the logarithm is \( -k \cdot \ln(4) \). Therefore, the expression \( \ln \left( \frac{1}{4^k} \right) \) is expanded as \( -k \cdot \ln(4) \).
Key Concepts
Power Rule of LogarithmsLogarithmic PropertiesExpressions Involving Logarithms
Power Rule of Logarithms
The power rule of logarithms is a fundamental property that makes logarithmic expressions much simpler to handle. This rule tells us how to manage expressions of the form \( \ln(a^b) \), where we are taking the logarithm of a number raised to a power. According to the power rule, \( \ln(a^b) = b \cdot \ln(a) \). This means that instead of keeping the exponent within the logarithm, we can bring it out as a multiplier. This is particularly helpful for expanding logarithmic expressions, as seen in the equation \( \ln(\frac{1}{4^k}) \).
Here, we recognize that \( \frac{1}{4^k} \) is the same as \( 4^{-k} \), meaning \( a = 4 \) and \( b = -k \). By applying the power rule, it becomes \( -k \cdot \ln(4) \). This expansion helps in simplifying and manipulating the expression, which is often useful in calculus and algebraic calculations.
Here, we recognize that \( \frac{1}{4^k} \) is the same as \( 4^{-k} \), meaning \( a = 4 \) and \( b = -k \). By applying the power rule, it becomes \( -k \cdot \ln(4) \). This expansion helps in simplifying and manipulating the expression, which is often useful in calculus and algebraic calculations.
Logarithmic Properties
Logarithmic properties are essential tools that help us transform complex logarithmic expressions into more manageable forms. These properties include the power rule, product rule, quotient rule, and change of base rule. Each serves a specific purpose for rearranging and breaking down expressions so that they're easier to solve or analyze.
Using these properties, you can rewrite a single logarithm into a sum, difference, or product of simpler logarithms. For example, when handling \( \ln(\frac{1}{4^k}) \), you apply the power rule to transform it into a product, namely \( -k \cdot \ln(4) \). Although this example didn't involve a sum or difference, the principle of expansion is based on the same set of properties that aid in simplifying expressions.
Using these properties, you can rewrite a single logarithm into a sum, difference, or product of simpler logarithms. For example, when handling \( \ln(\frac{1}{4^k}) \), you apply the power rule to transform it into a product, namely \( -k \cdot \ln(4) \). Although this example didn't involve a sum or difference, the principle of expansion is based on the same set of properties that aid in simplifying expressions.
Expressions Involving Logarithms
Expressions involving logarithms can often seem complicated, but using properties like the power rule can transform them into simpler forms. The goal is to expand or simplify these expressions to make them more understandable or easier to compute.
For instance, when you're given \( \ln(\frac{1}{4^k}) \), the task is to break it down using logarithmic properties. Instead of directly computing the logarithm of a fraction, you might first express it as a power, as \( 4^{-k} \). From here, applying the power rule of logarithms enables you to convert this into a straightforward product expression: \( -k \cdot \ln(4) \).
For instance, when you're given \( \ln(\frac{1}{4^k}) \), the task is to break it down using logarithmic properties. Instead of directly computing the logarithm of a fraction, you might first express it as a power, as \( 4^{-k} \). From here, applying the power rule of logarithms enables you to convert this into a straightforward product expression: \( -k \cdot \ln(4) \).
- Expansions can often lead to more insights, especially when solving equations or differentiating functions involving logarithms.
- This expression now reveals itself as a simple multiplication, which is often much easier to handle in further calculations.
Other exercises in this chapter
Problem 7
For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Find and interpret \(f(0)\). Round to the nearest tenth.
View solution Problem 7
For the following exercises, use like bases to solve the exponential equation. \(2^{-3 n} \cdot \frac{1}{4}=2^{n+2}\)
View solution Problem 7
For the following exercises, state the domain and range of the function. \(h(x)=\ln \left(\frac{1}{2}-x\right)\)
View solution Problem 7
For the following exercises, rewrite each equation in exponential form. \(\log _{a}(b)=c\)
View solution