Problem 45 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the expression as the logarithm of a single quantity. $$ 3[\ln x+\ln (x+3)-\ln (x+4)] $$

Step-by-Step Solution

Answer

The expression \(3[\ln x+\ln (x+3)-\ln (x+4)]\) simplifies to \( \ln \left( \frac{x^2+3x}{x+4} \right)^3 \)

Step 1: Reorganize the expression using the properties of logarithms

The expression can be reorganized using the properties of logarithms. The sums of logarithms can be expressed as the logarithm of a product, and the difference of logarithms can be expressed as the logarithm of a quotient, therefore, the given expression can be rewritten as follows: \[3 \cdot \ln \left( \frac{x(x+3)}{x+4} \right)\]

Step 2: Open the parentheses by multiplication

The expression inside the logarithm function \(\frac{x(x+3)}{x+4}\) can be simplified by performing the multiplication: \[3 \cdot \ln \left( \frac{x^2+3x}{x+4} \right)\]

Step 3: Apply the power rule of logarithms

The power rule for logarithms states that for any positive number \(a\), \(b\), and \(n\), \(\ln(b^n) = n \cdot ln(b)\). So using it to simplify the expression and get the following: \[\ln \left( \frac{x^2+3x}{x+4} \right)^3\]

Key Concepts

Properties of Logarithms

Logarithms simplify complex mathematical calculations. They transform multiplication into addition, division into subtraction, and exponentiation into multiplication, making them extremely useful when dealing with large numbers or solving complex equations. The fundamental properties of logarithms include:

• Product Property: The logarithm of a product is the sum of the logarithms.
• Quotient Property: The logarithm of a quotient is the difference of the logarithms.
• Power Property: The logarithm of a number raised to a power is the exponent multiplied by the logarithm of the number.

These properties allow for the rewriting of expressions involving logarithms in a more simplified and manageable form. They provide the foundation for understanding more complex log-based expressions.

Logarithm of a Product

The product property of logarithms is key to simplifying log expressions that involve multiplication. It states: \[ \log_b(MN) = \log_b(M) + \log_b(N) \]This means that if you have a logarithm of two multiplied terms, you can separate them into individual logarithms and add them together. It's incredibly useful for breaking down complicated expressions. For example, in the expression \( \ln(x(x+3)) \), you can rewrite it as:

• \( \ln(x) + \ln(x+3) \)

This transformation makes further simplification possible, especially when simplifying expressions with multiple logs.

Logarithm of a Quotient

Just as multiplication can become addition with logarithms, division becomes subtraction. The quotient property of logarithms is represented by the formula: \[ \log_b\left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \]When a logarithmic expression involves a division, it can be broken down into the difference of two logarithmic expressions. For example, in the expression \( \ln \left( \frac{x^2 + 3x}{x+4} \right) \), you simplify it by splitting the division into:

• \( \ln(x^2 + 3x) - \ln(x + 4) \)

This step is crucial when combining the logarithms into a single expression, as it aligns the approach with other logarithmic properties.

Power Rule of Logarithms

The power rule of logarithms helps in dealing with expressions where the variable is raised to a particular power. It states: \[ \log_b(M^n) = n\cdot \log_b(M) \]This rule allows you to move an exponent in front of a logarithm as a coefficient. It's particularly powerful when working with complex expressions that involve exponents. For example, in \( \ln \left( \left( \frac{x^2+3x}{x+4} \right)^3 \right) \), the exponent 3 can be moved in front of the logarithm:

• \( 3 \cdot \ln \left( \frac{x^2+3x}{x+4} \right) \)

By applying this property, you simplify the overall expression, making it easier for further manipulation or solving equations. It underlines the versatility and strength of logarithms in calculus and algebra.