Problem 20 - College Algebra with Corequisite Support

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

The expression condenses to \( \log(6x^9) \).

Step 1: Identify Logarithmic Property to Use

The expression consists of two logarithms being added: \( \log(2x^4) + \log(3x^5) \). When two logarithms with the same base are added, we can use the Product Property of logarithms, which states that \( \log_a(m) + \log_a(n) = \log_a(m \cdot n) \).

Step 2: Apply the Product Property

Using the Product Property of logarithms, we combine the two logarithmic expressions: \( \log(2x^4) + \log(3x^5) = \log((2x^4) \cdot (3x^5)) \).

Step 3: Simplify the Expression Inside the Logarithm

Multiply the expressions inside the logarithm: \((2x^4) \cdot (3x^5) = 6x^{4+5} = 6x^9 \). So, the expression simplifies to \( \log(6x^9) \).

Step 4: Write the Condensed Logarithmic Expression

Using the calculations, the condensed expression is \( \log(6x^9) \).

Key Concepts

Product Property of Logarithms

When working with logarithmic expressions, one of the key properties that often comes in handy is the Product Property of Logarithms. This property is useful when you are faced with the addition of two logarithms which have the same base.

• The Product Property states: \( \log_a(m) + \log_a(n) = \log_a(m \cdot n) \)
• This essentially means that when you have two logs being added, you can combine them into one log by multiplying the contents.

This property is a straightforward way to simplify complex expressions into more manageable forms. It's particularly useful in problems where you are required to condense logs, like the one in this exercise. Remember, it is applicable only when the bases of both logarithms are the same. Logarithms make it easier to handle large numbers or products, and this property is an illustration of their utility in mathematics.

Condensing Logarithms

Condensing logarithms involves the process of taking multiple logarithmic terms and combining them into a single expression. This is often done using various logarithmic properties, one of the most commonly used being the Product Property, as we've already discussed.

• The first step, in this case, is to identify that the expression consists of logarithms that can be combined.
• Next, apply the relevant property (Product in our example) to condense the terms.

For example, given the expression \( \log(2x^4) + \log(3x^5) \), we can condense this using the Product Property into \( \log((2x^4) \cdot (3x^5)) \). Remember, condensing is about transforming the expression into a singular log, making it easier to interpret or compute later on.

Logarithmic Expressions

Logarithmic expressions are statements that involve the logarithm of a number or variable. They are fundamental in mathematics for simplifying calculations involving exponential growth or decay.

• They can take many forms, such as products, quotients, and powers.
• In our exercise, the expression \( \log(2x^4) + \log(3x^5) \) involves logarithms of monomials.

Working with logarithmic expressions often means utilizing the properties of logarithms to simplify or manipulate them. The expressions can be combined, expanded, or condensed based on the context of the problem. In cases like ours, manipulating these expressions allows for easier interpretation and solution of problems.

Simplification of Expressions

Simplification is a crucial part of solving equations in algebra, and logarithmic expressions are no exception. Simplifying logarithmic expressions involves reducing them to their most straightforward form using mathematical properties, making them easier to handle or understand.

• It usually starts by applying properties like the Product Property to combine terms.
• Then, you multiply the terms inside the log, as shown in our example with \((2x^4) \cdot (3x^5) = 6x^{9}\).

The ultimate aim is to achieve a neat, concise expression like \( \log(6x^9) \), where previously complex equations have been distilled down. This simplification is helpful not only for solving mathematical problems but also for presenting results in the most digestible format.