Problem 37 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \ln(x) + \frac{1}{2}\ln(y) - \frac{1}{2}\ln(z) \)

Step 1: Identify the Components

We start by recognizing what components make up the expression \( \ln \left(x \sqrt{\frac{y}{z}}\right) \). Inside the logarithm, we have a multiplication: \( x \cdot \sqrt{\frac{y}{z}} \). Also, note that \( \sqrt{\frac{y}{z}} \) is equivalent to \((\frac{y}{z})^{1/2}\).

Step 2: Apply Logarithm Laws for Multiplication

The law \( \ln(ab) = \ln(a) + \ln(b) \) is used for separating products inside a logarithm into a sum. Apply it to get \( \ln(x \cdot \sqrt{\frac{y}{z}}) = \ln(x) + \ln\left(\sqrt{\frac{y}{z}}\right) \).

Step 3: Simplify the Square Root using Exponent Property

Recognize that \( \sqrt{\frac{y}{z}} = (\frac{y}{z})^{1/2} \, \). Thus, the logarithm \( \ln\left(\sqrt{\frac{y}{z}}\right) \) becomes \( \ln\left((\frac{y}{z})^{1/2}\right) \).

Step 4: Apply the Power Rule

Use the power rule of logarithms: \( \ln(a^b) = b \ln(a) \). Thus, \( \ln((\frac{y}{z})^{1/2}) = \frac{1}{2} \ln\left(\frac{y}{z}\right) \).

Step 5: Expand the Fraction with Logarithm Laws

Use the quotient rule of logarithms: \( \ln\left(\frac{y}{z}\right) = \ln(y) - \ln(z) \). So, \( \frac{1}{2} \ln\left(\frac{y}{z}\right) \) becomes \( \frac{1}{2}(\ln(y) - \ln(z)) \).

Step 6: Combine All Terms Together

Combine the results from previous steps to write the expression as \( \ln(x) + \frac{1}{2}\ln(y) - \frac{1}{2}\ln(z) \). This is the expanded form of the original expression using the laws of logarithms.

Key Concepts

Logarithm Properties

Logarithms are really useful to simplify complex expressions involving multiplication, division, and exponents.
The properties of logarithms help us break down these expressions so they are easier to understand and solve.
Some key properties of logarithms include:

• Product Rule: The logarithm of a product is the sum of the logarithms: \( \ln(ab) = \ln(a) + \ln(b) \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• Power Rule: The logarithm of an exponent can be moved as a multiplier: \( \ln(a^b) = b \ln(a) \).

By understanding these properties, you can expand, simplify, and solve logarithmic expressions efficiently.
This process allows expressions to be written in equivalently useful ways that make calculations and interpretations much simpler.

Exponent Rules

Exponents represent repeated multiplication, and they have specific rules that govern their behavior.
Understanding these rules is crucial for applying logarithmic properties correctly.Some common exponent rules include:

• Product of Powers: When multiplying like bases, add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• Quotient of Powers: When dividing like bases, subtract their exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• Zero Exponent: Any nonzero number raised to the power of zero is one: \( a^0 = 1 \).
• Negative Exponent: A negative exponent represents the reciprocal of the base raised to the opposite positive exponent: \( a^{-n} = \frac{1}{a^n} \).

Exponents are integral to working with roots, such as square roots, which can be expressed as exponents.
For instance, a square root can be rewritten as a power of 1/2: \( \sqrt{a} = a^{1/2} \).
This is vital knowledge for simplifying expressions with roots using logarithms.

Product and Quotient Rules

The product and quotient rules of logarithms are essential for breaking down expressions that involve multiplication and division inside a logarithm.
These rules allow you to write a single log expression in terms of multiple simpler logarithmic terms.

Product Rule

The product rule states that the logarithm of a product can be split into the sum of two separate logarithms.
For example, \( \ln(ab) = \ln(a) + \ln(b) \). This rule facilitates the expansion of terms inside a log.

Quotient Rule

Similarly, the quotient rule transforms the logarithm of a division into a difference of logs, making it easier to handle:
\( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
Remarkably, both rules reflect the thinking that multiplication becomes addition and division becomes subtraction inside logarithms.
This makes it straightforward to simplify expressions while preserving their mathematical meaning.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves using the laws of logarithms to spread out a complicated log term into a combination of simpler terms.
This can ultimately make solving equations and understanding expressions more manageable.

Steps to Expand Logarithmic Expressions

1. **Identify and separate products and quotients inside the log**: Use the product and quotient rules.
For example, transform \( \ln(x \sqrt{\frac{y}{z}}) \) into smaller parts: \( \ln(x) + \ln(\sqrt{\frac{y}{z}}) \).2. **Convert roots to exponents**: Recognize roots as fractional exponents.
In the example, \( \sqrt{\frac{y}{z}} \) becomes \( \left(\frac{y}{z}\right)^{1/2} \).3. **Apply the power rule**: Pull out exponents as coefficients of the log.
This makes \( \ln\left(\left(\frac{y}{z}\right)^{1/2}\right) \) become \( \frac{1}{2} \ln\left(\frac{y}{z}\right) \).4. **Finish up with the quotient rule**: Expand out the fraction using the quotient rule. This results in \( \frac{1}{2}(\ln(y) - \ln(z)) \).By following these steps systematically, you can expand any given logarithmic expression.
These expansions are potent tools in simplifying and solving logarithmic problems.