Problem 69 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log x+\log \left(x^{2}-1\right)-\log 7-\log (x+1) $$

Step-by-Step Solution

Answer

\(\log {(x^3-x)/(7x+7)}\)

Step 1: Apply the Logarithm Addition Rule

Use the product rule of logarithms, which states that the sum of two logarithms equals the logarithm of their product. Apply this rule to \(\log_x + \log {(x^2-1)}\): \[\log_x + \log {(x^2-1)} = \log {(x*(x^2-1))}\]

Step 2: Apply the Logarithm Subtraction Rule

Use the quotient rule of logarithms, which states that the difference of two logarithms equals the logarithm of the quotient of their values. Apply this rule to \(\log {(x*(x^2-1))} - \log 7 - \log (x+1)\): \[\log {(x*(x^2-1))} - \log 7 - \log (x+1) = \log {((x*(x^2-1))/(7*(x+1)))}\]

Step 3: Simplify the Expression

Simplify the expression as much as possible: \[\log {((x*(x^2-1))/(7*(x+1)))} = \log {(x^3-x)/(7x+7)}\]

Key Concepts

Properties of Logarithms

Logarithms have a set of useful properties that make managing complex expressions much easier. These properties help to simplify or condense expressions, providing key tools for solving algebraic problems involving logarithms. They stem from the definition of a logarithm, which is the inverse of an exponent. Logarithms turn multiplication into addition, division into subtraction, and exponentiation into multiplication, thanks to their unique properties.

• Product Rule helps combine two logarithms.
• Quotient Rule aids in simplifying log expressions.
• Power Rule manages powers within a log.

These tools make calculations involving large numbers or complex formulas manageable. Familiarity with these properties is essential for tackling equations working with logarithms efficiently.

Product Rule of Logarithms

The product rule is one of the fundamental properties of logarithms. It states that the logarithm of a product is equal to the sum of the logarithms of its factors. This is particularly handy when you need to simplify expressions where variables multiply.
For example, if you have two numbers, say `a` and `b`, the product rule says:
\(\log(a * b) = \log(a) + \log(b) \)
In the context of our exercise, the product rule combines \(\log x\) and \(\log(x^2-1)\) into one term. This allows you to transform terms into products, simplifying complex expressions into more workable forms for further operations.

Quotient Rule of Logarithms

The quotient rule for logarithms provides a method for simplifying differences between logarithmic expressions. It states that a logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator.In mathematical terms, for any positive numbers `a` and `b`, the quotient rule can be expressed as:
\(\log \left(\frac{a}{b}\right) = \log(a) - \log(b) \)
Within the exercise we're analyzing, this rule was applied after the product rule to combine several logarithmic expressions. Specifically, after using the product rule to condense the logarithms into a single term, the quotient rule further simplified the expression to a neater, single logarithm. This streamlined the initially cumbersome expression into a more manageable form, similar to how division simplifies multiplication into smaller numbers.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involving logarithms often involves condensing multiple terms into a single term, or breaking down complex expressions into simpler components. This process reduces the expression to its simplest form, making it easier to work with or understand.
Logarithms, with their properties, are particularly useful in simplification efforts. They can transform multiplication into addition, and division into subtraction, significantly reducing the complexity of many algebraic problems. For example, in our exercise, the expression \(\log {(x(x^2-1))}-\log 7-\log(x+1)\) was simplified into \(\log \left(\frac{x(x^2-1)}{7(x+1)}\right)\). Further simplification led to \(\log \left(\frac{x^3-x}{7x+7}\right)\).
These operations illustrate how effectively using logarithmic properties can simplify even daunting expressions into manageable forms, turning complex operations into straightforward tasks.