Problem 270 - Calculus Volume 1

Question

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. $$ \log x^{4} y $$

Step-by-Step Solution

Answer

\(\log(x^4 y) = 4 \log(x) + \log(y)\).

Step 1: Identify the Expression

The given expression is \( \log(x^4 y) \). We need to split this into a combination of the sum, difference, and product of logarithms.

Step 2: Apply the Product Property of Logarithms

The product property states \( \log(ab) = \log a + \log b \). Apply this to \( x^4 y \): \[ \log(x^4 y) = \log(x^4) + \log(y) \].

Step 3: Apply the Power Property of Logarithms

The power property states \( \log(a^b) = b \log a \). Apply this to \( \log(x^4) \): \[ \log(x^4) = 4 \log(x) \].

Step 4: Combine the Result

Combine the results from Steps 2 and 3: \[ \log(x^4 y) = 4 \log(x) + \log(y) \].

Key Concepts

Properties of Logarithms

Logarithms have several important properties that can help simplify the expression and manipulation of logarithmic terms. These properties are akin to the laws of arithmetic, making it easier to work with complex logarithmic expressions. Understanding these can be crucial for solving logarithmic equations effectively.

• The **Product** Property
• The **Quotient** Property
• The **Power** Property
• The **Change of Base** Formula

Each of these properties simplifies different aspects of logarithmic expressions involving products, quotients, and powers respectively. By breaking down expressions using these rules, more straightforward calculations can be achieved.
In the exercise presented, we specifically utilize the **Product** and **Power** properties to decompose the initial expression into a more manageable form.

Product Property of Logarithms

The Product Property of Logarithms is an invaluable tool for simplifying expressions. The rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. Specifically, it is written as:\[\log(a b) = \log(a) + \log(b)\]
In practice, this means when you have something like \( \log(x^4 y) \), you can separate it into two parts:

• \( \log(x^4) \)
• \( \log(y) \)

Using the product property, we've combined these two separate expressions into a single equation. This allows for further simplification or manipulation by applying additional properties if necessary.
Understanding this property can reduce the complexity of solving or interpreting logarithmic expressions and is foundational for working through more complex logarithms.

Power Property of Logarithms

The Power Property of Logarithms is another fundamental rule, which is incredibly useful when dealing with expressions that involve exponents. The power property can be expressed as:\[\log(a^b) = b \log(a)\]
This means that if you have a logarithm of a number raised to an exponent, you can move the exponent in front of the logarithm as a multiplier.
In our exercise, after applying the product property, we used the power property on \( \log(x^4) \) to rewrite it:

• \( \log(x^4) \) became \( 4\log(x) \)

This simplification is invaluable as it straightforwardly breaks down complex logarithmic expressions, allowing easier computation or further algebraic manipulations.
By mastering the power property, you can greatly enhance your ability to simplify and solve equations involving logarithms.