Problem 20 - College Algebra

Question

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. $$ \log \left(2 x^{4}\right)+\log \left(3 x^{5}\right) $$

Step-by-Step Solution

Answer

\( \log(6x^9) \)

Step 1: Apply the Product Property of Logarithms

The product property of logarithms states that \( \log(a) + \log(b) = \log(ab) \). Apply this property to the expression \( \log(2x^4) + \log(3x^5) \). So, \( \log(2x^4 \cdot 3x^5) \).

Step 2: Simplify the Expression Inside the Logarithm

Multiply the terms inside the logarithm: \( 2x^4 \cdot 3x^5 = 6x^{(4+5)} \). This simplifies to \( 6x^9 \).

Step 3: Write the Final Condensed Logarithm Expression

The expression \( \log(2x^4) + \log(3x^5) \) simplifies to \( \log(6x^9) \) after applying the product property and simplifying the expression inside the logarithm.

Key Concepts

Properties of Logarithms

There are several key properties of logarithms that make them powerful tools for simplifying complex expressions. Understanding these properties is crucial for working effectively with logarithmic expressions, especially when condensing or expanding logs. These properties include:

• Product Property: This property states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, it is given by \( \log(a) + \log(b) = \log(ab) \).
• Quotient Property: This states that the logarithm of a quotient is equal to the difference of the logarithms, represented as \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \).
• Power Property: This property indicates that the logarithm of a number raised to an exponent is the exponent times the logarithm of the base. It is expressed as \( \log(a^b) = b\log(a) \).

Each property can be applied in different scenarios to simplify expressions, allowing one to move swiftly from complex to simpler forms. Mastery of these properties is essential for anyone dealing with logarithmic calculations.

Product Property of Logarithms

The product property of logarithms is one of the most commonly used properties. It allows us to turn the sum of logarithms into the logarithm of a product, making expressions compact and easier to handle. This property states:

• \( \log(a) + \log(b) = \log(ab) \)

For example, consider the expression \( \log(2x^4) + \log(3x^5) \). Using the product property, we can combine the logarithms into a single log:
\( \log(2x^4 \cdot 3x^5) \).
This simplifies directly to \( \log(6x^9) \). The product property is particularly useful because it helps reduce clutter in expressions by condensing multiple logs into one. Applying this property requires multiplying the arguments (or inner terms) of the logs, as seen in this example.

Condensing Logarithms

Condensing logarithms is an important process in mathematics where multiple logarithmic expressions are combined into a single, more manageable expression. This can greatly simplify solving equations and evaluating complex expressions. The process involves using logarithmic properties effectively to reduce the number of logs.To condense the expression \( \log(2x^4) + \log(3x^5) \), you would:

• Apply the Product Property by multiplying the terms inside the logarithms.
• Combine as \( \log(2x^4 \cdot 3x^5) \) which simplifies inside to \( \log(6x^9) \).

Condensing logs is often a step in transforming equations into simpler forms for solving, as it allows one to work with fewer terms. This condensed form is easier to differentiate, integrate, or otherwise manipulate in various calculus and algebra problems.