Question

Write each expression as a single logarithm. \(\log \left(\frac{x^{2}+2 x-3}{x^{2}-4}\right)-\log \left(\frac{x^{2}+7 x+6}{x+2}\right)\)

Short answer

\(\text{log} \left( \frac{(x + 3)(x - 1)}{(x - 2)(x + 6)(x + 1)} \right)\).

Step-by-step solution

Identify the properties of logarithms

Use the property of logarithms that states \(\textbf{A - B = A/B}\) to combine the given logarithms. We have \(\text{log}(A) - \text{log}(B) = \text{log}(\frac{A}{B})\).

Apply the property to the given expression

Combine the logarithms under one log: \(\text{log} \left( \frac{\frac{x^2 + 2x - 3}{x^2 - 4}}{\frac{x^2 + 7x + 6}{x + 2}} \right)\).

Simplify the complex fraction

Simplify \(\frac{\frac{x^2 + 2x - 3}{x^2 - 4}}{\frac{x^2 + 7x + 6}{x + 2}}\) by multiplying the numerator by the reciprocal of the denominator: \(\frac{x^2 + 2x - 3}{x^2 - 4} \cdot \frac{x + 2}{x^2 + 7x + 6}\).

Factorize the polynomials

Factorize each polynomial: \(x^2 + 2x - 3 = (x + 3)(x - 1)\), \(x^2 - 4 = (x + 2)(x - 2)\), and \(x^2 + 7x + 6 = (x + 6)(x + 1)\).

Substitute the factors back into the expression

Substitute the factored forms: \(\frac{(x + 3)(x - 1)}{(x + 2)(x - 2)} \cdot \frac{x + 2}{(x + 6)(x + 1)}\).

Simplify the expression

Cancel out the common factors \(x + 2\) from the numerator and the denominator: \(\frac{(x + 3)(x - 1)}{(x - 2)(x + 6)(x + 1)}\).

Write the final answer

The simplified expression is \(\text{log} \left( \frac{(x + 3)(x - 1)}{(x - 2)(x + 6)(x + 1)} \right)\).

Key concepts

logarithm properties

Understanding the properties of logarithms is essential for solving log-based problems. One fundamental property is that the difference of two logs can be expressed as a single log. Specifically, \(\text{log}(A) - \text{log}(B) = \text{log}\bigg(\frac{A}{B}\bigg)\). This property allows you to combine multiple logarithms into one. When faced with expressions like \(\text{log}(A) - \text{log}(B)\), rewriting them as \(\text{log}\bigg(\frac{A}{B}\bigg)\) can significantly simplify the equation. Similarly, the sum of two logs can be written as a single log multiplication \(\text{log}(A) + \text{log}(B) = \text{log}(A \times B)\). These properties are vital tools for combining and simplifying logarithmic expressions. By leveraging these principles, you can transform and solve log equations more efficiently.

combining logarithms

Combining logarithms using their properties can streamline complex equations into simpler forms. For instance, consider the expression \(\text{log} \bigg(\frac{x^2 + 2x - 3}{x^2 - 4}\bigg) - \text{log} \bigg(\frac{x^2 + 7x + 6}{x + 2}\bigg)\). By applying the property \(\text{log}(A) - \text{log}(B) = \text{log}\bigg(\frac{A}{B}\bigg)\), you can combine these logs into one: \(\text{log} \bigg( \frac{\frac{x^2 + 2x - 3}{x^2 - 4}}{\frac{x^2 + 7x + 6}{x + 2}} \bigg)\). This simplification makes it easier to manipulate the equation further and solve for the unknowns. Remember, combining logs can greatly reduce the workload and make the solution process more straightforward.

factoring polynomials

Factoring polynomials is a critical skill in simplifying mathematical expressions. To factor a polynomial like \(x^2 + 2x - 3\), find two numbers that multiply to the constant term (-3) and add to the coefficient of the linear term (+2). The numbers +3 and -1 fit. Thus, \(x^2 + 2x - 3\) factors to \((x + 3)(x - 1)\). Similarly, the polynomial \(x^2 - 4\) is a difference of squares and factors to \((x + 2)(x - 2)\). For more complex polynomials like \(x^2 + 7x + 6\), identify numbers multiplying to +6 and adding to +7: +6 and +1. Therefore, \((x + 6)(x + 1)\). Factoring these polynomials helps in simplifying fractions and solving equations.

simplifying fractions

After factoring polynomials, you can simplify complex fractions effectively. Start by writing the fraction in terms of its factors. For example, transform \(\frac{\frac{x^2 + 2x - 3}{x^2 - 4}}{\frac{x^2 + 7x + 6}{x + 2}}\) into \(\frac{(x + 3)(x - 1)}{(x + 2)(x - 2)} \times \frac{x + 2}{(x + 6)(x + 1)}\). Next, cancel out any common factors appearing in both the numerator and the denominator. Here, \((x + 2)\) cancels out from both, leaving you with: \(\frac{(x + 3)(x - 1)}{(x - 2)(x + 6)(x + 1)}\). This simplification process is crucial for reducing the equation to its simplest form, making it easier to interpret and solve. Always look for common terms to cancel, and ensure all polynomials are fully factored to avoid missing any simplifications.