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. $\log x + \log(x^2 - 1) - \log 7 - \log(x + 1)$

Verified step by step guidance

1. Identify the properties of logarithms that will help condense the expression: the product rule \(\log a + \log b = \log(ab)\) and the quotient rule \(\log a - \log b = \log\left(\frac{a}{b}\right)\).
2. Apply the product rule to combine the positive logarithms: \(\log x + \log(x^2 - 1) = \log\left(x(x^2 - 1)\right)\).
3. Apply the quotient rule to combine the negative logarithms: \(- \log 7 - \log(x + 1) = - \log(7(x + 1)) = \log\left(\frac{1}{7(x + 1)}\right)\).
4. Combine the results from steps 2 and 3 using the quotient rule: \(\log\left(x(x^2 - 1)\right) + \log\left(\frac{1}{7(x + 1)}\right) = \log\left(\frac{x(x^2 - 1)}{7(x + 1)}\right)\).
5. Recognize that \(x^2 - 1\) is a difference of squares and factor it as \((x - 1)(x + 1)\), then simplify the expression inside the logarithm by canceling common factors.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product rule (log a + log b = log(ab)), the quotient rule (log a - log b = log(a/b)), and the power rule (k log a = log(a^k)). These allow combining or breaking down logarithmic expressions to simplify or condense them.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves factoring and reducing expressions inside the logarithms. For example, recognizing that x^2 - 1 factors as (x - 1)(x + 1) helps in canceling terms when combined with other logarithms, making the expression easier to condense.

Evaluating Logarithmic Expressions Without a Calculator

Evaluating logarithmic expressions without a calculator requires recognizing values that simplify to known logarithms, such as log 1 = 0 or log of perfect powers. This skill helps in simplifying the final expression or determining if it can be further reduced to a numerical value.