Problem 270 - Calculus Volume 1

Question

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

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

Step 1: Apply the Product Rule

We start by applying the product rule for logarithms, which states that the logarithm of a product is the sum of the logarithms: \( \log(a \cdot b) = \log a + \log b \). For the expression \( \log x^4 y \), we can separate it as \( \log x^4 + \log y \).

Step 2: Apply the Power Rule

Next, we use the power rule of logarithms, which states that \( \log(a^b) = b \cdot \log a \). In the term \( \log x^4 \), we apply this rule to obtain \( 4 \cdot \log x \).

Step 3: Combine the Results

Combine the results from the previous steps to write the expression as a sum: \( 4 \cdot \log x + \log y \).

Key Concepts

Product Rule

Logarithms are a fascinating concept that simplify complex multiplications into manageable sums. One of the key properties in logarithms is the **Product Rule**. Imagine trying to turn a complex multiplication inside a log, like \( \log(a \cdot b) \), into a simpler form. The product rule helps us do exactly that.

• Definition: The product rule states that the logarithm of a product is equal to the sum of the logarithms of the factors.
• Formula: \( \log(a \cdot b) = \log a + \log b \)

To apply the product rule, you simply separate the factors of the product into individual log expressions. For instance, in the expression \( \log x^4 y \), you can break it down using this rule:\[\log x^4 y = \log x^4 + \log y\]This is a foundational rule that allows us to transform and simplify logarithmic expressions involving products. It helps break down more complex equations and makes calculations more feasible, especially when dealing with large numbers or variables.

Power Rule

The **Power Rule** is another essential rule that comes handy when working with logarithms. When you are dealing with a logarithm of a number raised to a power, the power rule provides a neat way to simplify the expression.

• Definition: The power rule states that the logarithm of a number raised to an exponent is equivalent to the exponent multiplied by the logarithm of the base.
• Formula: \( \log(a^b) = b \cdot \log a \)

Picture the expression \( \log x^4 \). Using the power rule, we can simplify it by moving the power 4 in front of the logarithm:\[\log x^4 = 4 \cdot \log x\]This rule not only simplifies expressions but also helps in solving equations where powers are involved. By handling exponents outside the logarithm, the algebra becomes more straightforward. This transformation is especially helpful in problems that require disentangling components of logarithmic functions.

Logarithms

Logarithms, often abbreviated as "logs," are the inverse operations of exponentiation. They provide a way to figure out how many times one number (the base) is multiplied to get another number.

• Definition: The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.
• Notation: \( \log_b(x) \) typically expresses logarithms where \( b \) is the base and \( x \) is the number.

Logs convert challenging multiplicative relationships into simpler additive ones. This is why properties like the product and power rules are so powerful. They allow for simplification and evaluation of complex expressions.Understanding logarithms is key in a range of mathematical topics and real-world applications, from calculating interest rates to deciphering earthquake magnitudes. Logs help us manipulate and understand exponential growth and decay processes. Mastering the basics and properties of logarithms, like product and power rules, makes tackling these concepts less intimidating and more approachable.