Problem 50 - Algebra and Trigonometry

Question

\(45-54\) . Use the Laws of Logarithms to combine the expression. $$ \ln (a+b)+\ln (a-b)-2 \ln c $$

Step-by-Step Solution

Answer

\( \ln(\frac{(a+b)(a-b)}{c^2}) \)

Step 1: Apply the Product Rule

The Product Rule of logarithms states that \( \ln x + \ln y = \ln(xy) \). Here, apply it to combine the first two terms: \( \ln(a+b) + \ln(a-b) = \ln((a+b)(a-b)) \).

Step 2: Use the Quotient Rule

The Quotient Rule of logarithms states that \( \ln x - \ln y = \ln(\frac{x}{y}) \). Apply this rule to the combined expression from Step 1 and the remaining term: \( \ln((a+b)(a-b)) - 2\ln c = \ln(\frac{(a+b)(a-b)}{c^2}) \), since \( 2\ln c \) means \( \ln(c^2) \).

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a fundamental property that allows us to simplify the sum of two logarithms into a single logarithm.
If you have an expression such as \( \ln x + \ln y \), this can be rewritten using the product of \( x \) and \( y \) inside a single logarithm: \( \ln(xy) \).
This rule is particularly useful in reducing the number of terms in logarithmic expressions and makes computations easier.
For example, in the given exercise, combining \( \ln(a+b) \) and \( \ln(a-b) \) into \( \ln((a+b)(a-b)) \) simplifies the expression significantly.
Remember, the product rule only applies when the base of the logarithms is the same, so ensure this consistency in your work.

• Useful for converting a sum of logs into a single log expression.
• Makes expressions shorter and often more manageable.
• Always check that logarithms have the same base before applying the rule.

Quotient Rule of Logarithms

Another key property is the quotient rule, which helps in simplifying a difference between two logarithms.
This rule is defined as \( \ln x - \ln y = \ln\left(\frac{x}{y}\right) \).
By using the quotient rule, you replace a subtraction of logarithms with a division inside a single logarithm.
In the context of our example, we apply the quotient rule to \( \ln((a+b)(a-b)) - 2\ln c \), simplifying it to \( \ln\left(\frac{(a+b)(a-b)}{c^2}\right) \).
This application also involves recognizing that \( 2\ln c \) can be rewritten as \( \ln(c^2) \).

• Converts subtraction of logs into a division within a log.
• Useful for streamlining expressions that involve subtraction of logs.
• Ensure to convert multiplicative constants appropriately, for instance, using rules like \( n\ln b = \ln(b^n) \).

Properties of Logarithms

Understanding the properties of logarithms is essential for simplifying complex expressions.
These properties, including the product and quotient rules, along with the power rule, help with various algebraic manipulations.
The core properties include:

• Product Rule: \( \ln x + \ln y = \ln(xy) \)
• Quotient Rule: \( \ln x - \ln y = \ln\left(\frac{x}{y}\right) \)
• Power Rule: \( n\ln x = \ln(x^n) \)
• Log of One: \( \ln 1 = 0 \)
• Base Rule: If \( \log_b(b^x) = x \), meaning the log base is the same as the exponential base, the result is the exponent.

These properties streamline the process of solving logarithm-related problems by converting expressions into simpler, more manageable forms.
Being familiar with these can save time and improve accuracy in calculation significantly.