Problem 42 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \sqrt{x \sqrt{y} \sqrt{z}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \log(x) + \frac{1}{4} \log(y) + \frac{1}{4} \log(z) \)

Step 1: Apply the Power Rule for Logarithms

The first step in expanding the logarithmic expression is to identify components that can have powers applied using the Power Rule for logarithms. Recall that the Power Rule states \( \log(a^b) = b \cdot \log(a) \). The entire expression is inside a square root, so it's equivalent to raising the expression to the power of \( \frac{1}{2} \). Thus, we have: \[ \log \sqrt{x \sqrt{y} \sqrt{z}} = \frac{1}{2} \cdot \log(x \sqrt{y} \sqrt{z}) \]

Step 2: Expand the Expression Inside the Logarithm

The next step is to consider how the expression inside the logarithm can be further expanded. We have \( x \sqrt{y} \sqrt{z} \), which can be rewritten using exponents: \( x \cdot y^{\frac{1}{2}} \cdot z^{\frac{1}{2}} \). So, we can use the properties of logarithms to separate the product: \[ \frac{1}{2} \cdot \log(x \cdot y^{\frac{1}{2}} \cdot z^{\frac{1}{2}}) = \frac{1}{2} \cdot (\log(x) + \log(y^{\frac{1}{2}}) + \log(z^{\frac{1}{2}})) \]

Step 3: Apply the Power Rule to Each Logarithm

Now that we have expanded the logarithm based on multiplication, apply the Power Rule to each term involving a power. This means taking the exponent in each \( \log(y^{\frac{1}{2}}) \) and \( \log(z^{\frac{1}{2}}) \) outside of the logarithm: \[ \frac{1}{2} \cdot (\log(x) + \frac{1}{2} \cdot \log(y) + \frac{1}{2} \cdot \log(z)) \]

Step 4: Distribute the Coefficient

The final step is to distribute the factor \( \frac{1}{2} \) across each term within the parentheses, resulting in the fully expanded expression: \[ \frac{1}{2} \cdot \log(x) + \frac{1}{4} \cdot \log(y) + \frac{1}{4} \cdot \log(z) \]

Key Concepts

Laws of Logarithms

The Laws of Logarithms are fundamental rules that simplify expressions involving logarithms. These laws help in breaking down complex logarithmic expressions into simpler parts, making them easier to manage and solve. The three key laws are:

• Product Law: This law states that the logarithm of a product is the sum of the logarithms of the individual factors. Mathematically expressed as: \( \log(a \cdot b) = \log(a) + \log(b) \).
• Quotient Law: The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).
• Power Law: This states that the logarithm of an exponentiated number is the exponent times the logarithm of the base: \( \log(a^b) = b \cdot \log(a) \).

These laws allow us to expand, condense, and transform logarithmic expressions into more manageable forms. They are particularly useful in solving equations and simplifying complex expressions.

Power Rule for Logarithms

The Power Rule is one of the Laws of Logarithms and is immensely helpful when dealing with exponents inside a logarithmic expression. The Power Rule states: \( \log(a^b) = b \cdot \log(a) \). This means that if you have a logarithm containing a base raised to an exponent, you can "bring down" the exponent to be a multiplier in front of the logarithm.

Let's take an example to understand this rule better. Suppose you have \( \log(8^3) \). By applying the power rule, this becomes \( 3 \cdot \log(8) \). Essentially, it turns a potentially complex problem into simple multiplication, thus making it easier to handle.This rule is particularly useful when you deal with roots, as roots can be represented as fractional exponents. For example, the square root of \( x \), or \( \sqrt{x} \), can be rewritten as \( x^{1/2} \), allowing the use of the Power Rule effectively.

Logarithmic Properties

Logarithmic properties extend beyond the basic laws and incorporate other features that make them versatile in various mathematical tasks. These properties include the ability to change the base of a logarithm and the inversion of operations.

• Change of Base Formula: Sometimes you might need to change the base of a logarithm to make calculations easier or conform to specific requirements. The formula is: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), where \( k \) is the new base.
• Inverse Property: Exponents and logarithms are inverse operations. That is, \( b^{\log_b(a)} = a \) and \( \log_b(b^a) = a \). This property helps in solving exponential equations using logarithms.

These properties are invaluable when simplifying expressions, solving logarithmic equations, or converting logarithms to different bases.

Exponentiation in Logarithms

Exponentiation is a key concept in understanding logarithms, as logarithms are essentially the inverse operation of exponentiation. Knowing how exponentiation works inside a logarithmic expression can greatly enhance your skills in expanding and simplifying these expressions.When you encounter a logarithm of an expression involving exponents, it's crucial to know how to manipulate and expand these terms. For instance, given a logarithm of a product of power terms, like \( \log(x^a \cdot y^b) \), you can rewrite it using the logarithmic laws as \( a \cdot \log(x) + b \cdot \log(y) \). This is essentially applying both the Product Law and the Power Rule at the same time.Understanding the relationship between exponentiation and logarithms allows for efficient handling of logarithmic equations and expressions. It simplifies the complexity of terms by breaking them into parts that adhere to simpler rules and operations, thereby fostering better clarity and comprehension.