Problem 48 - Essential Calculus Early Transcendentals

Question

Use the properties of logarithms to expand the quantity. $$ \log _{10} \sqrt{\frac{x-1}{x+1}} $$

Step-by-Step Solution

Answer

\(\frac{1}{2} \cdot \log_{10}(x-1) - \frac{1}{2} \cdot \log_{10}(x+1)\)

Step 1: Express the Square Root as a Power

First, recognize that a square root can be expressed as a power. So, rewrite the expression inside the logarithm as:\[\log_{10} \left( \frac{x-1}{x+1} \right)^{1/2}\]

Step 2: Apply the Power Rule of Logarithms

Using the power rule for logarithms, which states \(\log_b(a^n) = n \cdot \log_b(a)\), apply it to the expression:\[\log_{10} \left(\frac{x-1}{x+1}\right)^{1/2} = \frac{1}{2} \cdot \log_{10} \left( \frac{x-1}{x+1} \right)\]

Step 3: Use the Quotient Rule of Logarithms

The quotient rule for logarithms says \(\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\). Apply this to the expression inside the logarithm:\[\frac{1}{2} \cdot \left( \log_{10}(x-1) - \log_{10}(x+1) \right)\]

Step 4: Simplifying the Expression

Now distribute the \(\frac{1}{2}\) across the entire expression:\[\frac{1}{2} \cdot \log_{10}(x-1) - \frac{1}{2} \cdot \log_{10}(x+1)\]

Key Concepts

Properties of Logarithms

Logarithms have several important properties that make manipulating them straightforward and predictable. These properties allow us to expand, compress, and simplify logarithmic expressions. The key properties include:

• Product Rule
• Quotient Rule
• Power Rule
• Change of Base Formula

These properties are derived from the corresponding properties of exponents. For example, the product rule for logarithms is similar to the addition of exponents. Understanding these properties is crucial when dealing with complex expressions involving logs, as they provide the tools needed for expansion or simplification.
In our case, we primarily focus on the Power Rule and the Quotient Rule to expand the logarithmic expression. These rules help break down the expression into simpler, more manageable parts, making it easier to work with.

Power Rule

The Power Rule of logarithms is an incredibly useful tool when you have an exponent inside a logarithmic expression. It is expressed as follows:\[ \log_b(a^n) = n \cdot \log_b(a) \]This rule states that you can "bring down" the exponent in front of the logarithm as a multiplier. This is particularly handy for expressions where raising a term to a power is involved. Using this rule can significantly simplify the computation process, especially when dealing with roots or powers.
A common application of the Power Rule is when you have a square root, which can be rewritten as a power of one-half. In the given exercise, the expression \( \log_{10} \sqrt{\frac{x-1}{x+1}} \) is rewritten in terms of powers, allowing for an easier expansion. By applying the Power Rule, you can move that \( \frac{1}{2} \) out front, paving the way for further simplification using other logarithmic rules.

Quotient Rule

The Quotient Rule for logarithms helps us to handle division inside a logarithmic term. It is expressed as:\[ \log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c) \]This rule allows us to separate the logarithm of a quotient into the difference of two logarithms. It's particularly useful when you're looking to deconstruct a complex ratio into more digestible pieces.
In our exercise, after applying the Power Rule, we apply the Quotient Rule to the inner expression \( \log_{10} \left( \frac{x-1}{x+1} \right) \). This expands into two separate logs, \( \log_{10}(x-1) \) and \( \log_{10}(x+1) \), subtracted from each other. This transformation into a difference simplifies further operations and clarifies the structure of the expression. Now, each part can be handled independently, demonstrating the power of logarithmic rules in simplifying and solving complex expressions.