Problem 31 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to expand the expression. $$\ln \sqrt{a b}$$

Step-by-Step Solution

Answer

\( \ln \sqrt{ab} = \frac{1}{2} \ln(a) + \frac{1}{2} \ln(b) \).

Step 1: Express the Square Root as a Fractional Exponent

The square root of any expression can be represented as a fractional exponent. Therefore, express \( \sqrt{ab} \) as \( (ab)^{1/2} \). So, \( \ln \sqrt{ab} \) becomes \( \ln (ab)^{1/2} \).

Step 2: Apply the Power Rule of Logarithms

The Power Rule of Logarithms states that \( \ln(x^n) = n \cdot \ln(x) \). Apply this to \( \ln (ab)^{1/2} \) to obtain \( \frac{1}{2} \ln(ab) \).

Step 3: Use the Product Rule of Logarithms

The Product Rule for Logarithms is \( \ln(xy) = \ln(x) + \ln(y) \). Apply this rule to \( \ln(ab) \), giving \( \ln(a) + \ln(b) \).

Step 4: Combine All Steps

Substitute the result from Step 3 back into the result from Step 2. We have \( \frac{1}{2} (\ln(a) + \ln(b)) = \frac{1}{2} \ln(a) + \frac{1}{2} \ln(b) \).

Key Concepts

Power Rule of Logarithms

The Power Rule of Logarithms is a handy tool in simplifying logarithmic expressions. It tells us how to handle logarithms involving exponents. This rule states that for any positive number \( x \) and any real number \( n \), the logarithm of \( x^n \) can be rewritten as \( n \times \ln(x) \).
Imagine you are faced with \( \ln((ab)^{1/2}) \), like in our original problem. By using the Power Rule, we can move the exponential factor (in this case, \( 1/2 \)) in front of the logarithm. This transforms the expression into \( \frac{1}{2} \ln(ab) \).
This not only simplifies the expression but makes further manipulation easier. Remember, knowing this rule can save you time and effort when dealing with complex expressions.

Product Rule of Logarithms

The Product Rule of Logarithms allows us to break down the logarithm of a product into a sum of individual logarithms. The rule states that for any positive numbers \( x \) and \( y \), \( \ln(xy) = \ln(x) + \ln(y) \).
This principle is especially useful when you want to simplify expressions like \( \ln(ab) \), breaks it into components: \( \ln(a) + \ln(b) \). By doing this, we transform complicated expressions into simpler parts, each dealt with individually.
Embracing the Product Rule makes it easier to handle multiplication inside a logarithm, much like factoring in algebra. It showcases how multiplication compresses information, and logarithms help us unravel it.

Logarithmic Expansion

Logarithmic Expansion involves using the rules of logarithms to stretch and open up a logarithmic expression, making it simpler or more revealing. At its heart, this process converts a compact logarithmic form into an expansive sum or difference.
In our example, we started with \( \ln \sqrt{ab} \). By applying the Power and Product Rules, we expanded this into \( \frac{1}{2} \ln(a) + \frac{1}{2} \ln(b) \).
This expansive form can be beneficial for solving equations, simplifying calculation, or even revealing patterns or relationships within data. Mastery of logarithmic expansion allows you to manipulate and understand expressions at a more fundamental level, much like unfolding a map to see all its parts.