Problem 47 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the expression as the logarithm of a single quantity. $$ \frac{3}{2}\left[\ln x\left(x^{2}+1\right)-\ln (x+1)\right] $$

Step-by-Step Solution

Answer

The simplified logarithm of a single quantity is \( \ln \left( x (x^{2} + 1) \right) \)

Step 1: Handle the Coefficient Inside the Parenthesis

Using the property of logarithms \(\ln a^n = n \ln a\), we can pull the constant \(\frac{3}{2}\) inside the logarithm as an exponent. Hence we write, \( \frac{3}{2} \ln x (x^2 + 1) - \frac{3}{2} \ln (x + 1) \) as \( \ln x^{\frac{3}{2}} (x^2 + 1)^{\frac{3}{2}} - \ln (x + 1)^{\frac{3}{2}} \).

Step 2: Combine the Logs

Next, we can combine the two separate logarithms into one using the quotient rule for logarithms. The quotient rule states that \(\log_b⁡ {a} - \log_b⁡ {c} = \log_b⁡ {a/c}\). This implies that our new expression becomes \( \ln \left( \frac{x^{\frac{3}{2}} (x^2 + 1)^{\frac{3}{2}}}{(x + 1)^\frac{3}{2}} \right) \)

Step 3: Simplify the result

The final step is to simplify the expression inside the logarithm to get the answer in the simplified form. After simplifying we get \( \ln \left( x (x^{2} + 1) \right) \).

Key Concepts

Logarithmic Rules

Logarithmic rules are the backbone of simplifying and manipulating logarithmic expressions. They offer us techniques to convert complex logarithmic forms into more manageable expressions. Here are a few key rules:

• Product Rule: This rule combines logarithms of products. It states that \( \log_b(a) + \log_b(c) = \log_b(ac) \).
• Quotient Rule: Used to combine logarithms of quotients, this rule says \( \log_b(a) - \log_b(c) = \log_b(\frac{a}{c}) \).
• Power Rule: This rule allows exponents in a logarithmic term to be moved outside as multipliers: \( \log_b(a^n) = n \log_b(a) \).

Understanding these rules is crucial as they provide simplification pathways for complex logarithmic expressions. These rules can turn a daunting equation into a much simpler one, especially when combined effectively. Below, we'll explore how these rules can specifically help in reducing the complexity of logarithmic expressions, such as those found in mathematical exercises or textbook problems.

Simplifying Logarithms

Simplifying logarithms involves using logarithmic rules to reduce expressions to their simplest form. This process often includes applying the rules like product, quotient, and power rules. For example, consider the task of simplifying an expression starting with \( \frac{3}{2} \ln \left( x(x^2+1) \right) - \frac{3}{2} \ln (x+1) \).
Using the power rule, we can move the coefficient \( \frac{3}{2} \) to the power of the arguments inside the logarithms, now expressing it as \( \ln \left( x^{\frac{3}{2}}(x^2+1)^{\frac{3}{2}} \right) - \ln (x+1)^{\frac{3}{2}} \).
Next, by the quotient rule, these can be combined into a single logarithm as \( \ln \left( \frac{x^{\frac{3}{2}}(x^2+1)^{\frac{3}{2}}}{(x+1)^{\frac{3}{2}}} \right) \). This step embodies simplification by using logarithmic identities. Further simplification, often involving algebraic manipulation, can reduce this expression down to \( \ln \left( x(x^2+1) \right) \).The final expression is cleaner and easier to interpret, showcasing how logarithmic rules can effectively simplify to help solve problems efficiently.

Properties of Logarithms

The properties of logarithms are fundamental truths used in their manipulation and transformation. These properties are essential for solving equations involving logarithms, making complex problems approachable. Here are a few important properties:

• One-to-One Property: If \( \log_b(a) = \log_b(c) \), then \( a = c \). This property is useful in solving equations where two expressions in their logarithmic form are equal.
• Change of Base Formula: If you need to switch the base of a logarithm, use \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \) to convert bases.
• Logarithmic Identity: The identity \( \log_b(b) = 1 \) for any base \( b \) is fundamental when solving for unknowns in logarithmic equations.

These properties are not just theoretical but serve practical purposes. They are applied to solve logarithmic equations, transform expressions, and even in calculus for differentiating or integrating logarithmic functions. Embracing these properties allows for fluid maneuvering through different kinds of logarithmic challenges and paves the way for more advanced mathematical explorations. In summary, by understanding and utilizing these properties, students can confidently and accurately work through even the most daunting logarithmic expressions.