Problem 39 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{3 x(x+1)}{(2 x+1)^{2}} $$

Step-by-Step Solution

Answer

The expression in terms of sum, difference, or multiple of logarithms is \( \ln 3 + \ln x + \ln(x+1) - 2*\ln(2x+1) \)

Step 1: Apply the Quotient Rule

Use the quotient rule of logarithms to split the log of a quotient into a difference of logs. The quotient rule says that \( \ln \frac{a}{b} = \ln(a) - \ln(b) \). Hence, apply the quotient rule to \( \ln \frac{3 x(x+1)}{(2 x+1)^{2}} \). This gives: \( \ln 3x(x+1) - \ln (2x+1)^2 \)

Step 2: Apply the Product Rule

Use the product rule to split the log of a product into a sum of logs. The product rule says that \( \ln(ab) = \ln(a) + \ln(b) \). Apply the product rule to \( \ln 3x(x+1) \). This gives: \( \ln 3 + \ln x + \ln(x+1) \). So our expression is now \( \ln 3 + \ln x + \ln(x+1) - \ln (2x+1)^2 \)

Step 3: Apply the Power Rule

Use the power rule of logarithms to bring down the power as a multiple. The power rule says that \( \ln a^b = b*\ln a \). Apply the power rule to \( \ln (2x+1)^2 \). This gives: \( 2*\ln(2x+1) \). Replacing in our expression gives us \( \ln 3 + \ln x + \ln(x+1) - 2*\ln(2x+1) \)

Key Concepts

Quotient Rule

Logarithms offer an elegant way to handle division within expressions using the quotient rule. This rule helps us transform the logarithm of a quotient into the subtraction of two separate logarithms. The quotient rule states: \[ \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \] This means that when you are faced with a logarithm of a fraction, you can express it as the difference between two logs. For example, in the given exercise, we transformed \( \ln \frac{3x(x+1)}{(2x+1)^2} \) into \( \ln 3x(x+1) - \ln (2x+1)^2 \). This property is particularly helpful in simplifying complex logarithmic expressions, allowing you to isolate terms and work with them individually. By breaking a fraction into its components, each part can be further simplified using other logarithmic properties.

Product Rule

The product rule is a fundamental property of logarithms that simplifies expressions containing multiplication into addition. It states: \[ \ln(ab) = \ln(a) + \ln(b) \] This allows us to split a single logarithm of a product into the sum of logarithms of its factors. In the context of our example, we applied the product rule to \( \ln 3x(x+1) \) to break it down further into \( \ln 3 + \ln x + \ln(x+1) \). This rule is incredibly useful in both simplifying expressions and solving logarithmic equations. By expressing multiplication as addition, calculations become more manageable as it breaks down the original expressions into easier parts. This helps us tackle each segment step by step, leading to more straightforward solutions.

Power Rule

The power rule is a vital property that deals with exponents within logarithmic expressions by transforming powers into coefficients. It follows this principle: \[ \ln(a^b) = b \times \ln(a) \] This rule allows you to "bring down" the power to become a multiplier before the logarithm, making expressions simpler. In step 3 of our solution, we used the power rule on \( \ln (2x+1)^2 \) to rewrite it as \( 2 \times \ln(2x+1) \). By utilizing the power rule, even complex expressions with high powers become easier to manage. Remember, any time you see an exponent inside a logarithmic expression, this rule will help you shift that exponent outside as a factor, reducing the complexity and making further simplifications easier.