Problem 36 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{5} \sqrt{\frac{x-1}{x+1}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \log _{5}(x-1) - \frac{1}{2} \log _{5}(x+1) \)

Step 1: Identify the Original Expression

The expression given is \( \log _{5} \sqrt{\frac{x-1}{x+1}} \). This can initially be rewritten in terms of exponents: \( \log _{5} \left( \frac{x-1}{x+1} \right)^{1/2} \).

Step 2: Apply the Power Rule

According to the power rule of logarithms, \( \log_b(x^a) = a \cdot \log_b(x) \). Apply this to the expression to get: \( \frac{1}{2} \log _{5} \left( \frac{x-1}{x+1} \right) \).

Step 3: Apply the Quotient Rule

The quotient rule states that \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). Using this, the expression becomes \( \frac{1}{2} \left( \log _{5}(x-1) - \log _{5}(x+1) \right) \).

Step 4: Distribute the Constant

Distribute the \( \frac{1}{2} \) across the terms in the parentheses: \( \frac{1}{2} \log _{5}(x-1) - \frac{1}{2} \log _{5}(x+1) \).

Step 5: Final Expanded Form

The expression is now fully expanded into the form: \( \frac{1}{2} \log _{5}(x-1) - \frac{1}{2} \log _{5}(x+1) \).

Key Concepts

Laws of Logarithms

The laws of logarithms are essential rules that simplify the complexity of logarithmic expressions and computations. Understanding these laws allows us to break down and manipulate logarithmic expressions into simpler components. Here are the three primary laws of logarithms:\

• Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of its factors: \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \).
• Quotient Rule: It indicates that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).
• Power Rule: This rule expresses the logarithm of a power as the product of the exponent and the logarithm of the base: \( \log_b\left(m^a\right) = a \cdot \log_b(m) \).

These laws are highly applicable when expanding logarithmic expressions, like the problem above, where you systematically use the rules for simplification.

Power Rule of Logarithms

The power rule of logarithms is a powerful tool for handling expressions where numbers are raised to exponents inside a logarithm. It helps to simplify the logarithm by transforming the exponent into a multiplier.Let's see how this works in practice:

• For any logarithm in the form \( \log_b(x^a) \), you can rewrite it by bringing the exponent \( a \) out in front, leading to the simplified expression \( a \cdot \log_b(x) \).

In the provided exercise, the original expression \( \log_{5} \sqrt{\frac{x-1}{x+1}} \) was rewritten as \( \log_{5} \left(\frac{x-1}{x+1}\right)^{1/2} \). Using the power rule, this becomes \( \frac{1}{2} \cdot \log_{5} \left(\frac{x-1}{x+1}\right) \). The power rule significantly helps to simplify complex logarithmic relationships, enabling easier manipulation and further expansion.

Quotient Rule of Logarithms

The quotient rule of logarithms is essential when dealing with expressions that involve division inside a log function. It transforms a single logarithm into a subtraction of two separate logarithms.Here's a closer look at this rule:

• For any expression in the form \( \log_b\left(\frac{m}{n}\right) \), it can be broken into two parts: \( \log_b(m) - \log_b(n) \).

In the given exercise, after applying the power rule, the expression \( \frac{1}{2} \cdot \log_{5} \left(\frac{x-1}{x+1}\right) \) can be expanded using the quotient rule as \( \frac{1}{2} \cdot (\log_{5}(x-1) - \log_{5}(x+1)) \). This splits the logarithm of a fraction into the logarithm of the numerator minus the logarithm of the denominator, which simplifies understanding and manipulation.

Expanding Expressions

Expanding expressions in logarithmic form means transforming a complex log expression into a sum, difference, or multiple single logs, which are much easier to understand and compute.Let's expand on the importance of this process:

• Expanding involves applying the laws of logarithms iteratively to dismantle a composite log expression into simpler, smaller parts.
• The main purpose is to evaluate or simplify logarithmic expressions further, making them more tractable or suitable for solving equations.
• For problems like our exercise, expansion simplifies a complex initial expression into more straightforward terms: \( \frac{1}{2} \log_{5}(x-1) - \frac{1}{2} \log_{5}(x+1) \).

By expanding, we convert a challenging problem into manageable steps using rules we've learned, which enhances understanding and accuracy.