In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log₇ (7/x)

Verified step by step guidance

1. Recall the logarithmic property that states \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). This allows us to rewrite the logarithm of a quotient as the difference of two logarithms.
2. Apply this property to the expression \( \log_7 \left( \frac{7}{x} \right) \), which gives \( \log_7 7 - \log_7 x \).
3. Recognize that \( \log_7 7 \) asks the question: "To what power must 7 be raised to get 7?" Since 7 raised to the power 1 is 7, \( \log_7 7 = 1 \).
4. Substitute this value back into the expression to get \( 1 - \log_7 x \).
5. The expression is now fully expanded and simplified as much as possible without a calculator.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions in simpler or expanded forms. For example, the quotient rule states that log_b(M/N) = log_b(M) - log_b(N), which is essential for expanding expressions like log7(7/x).

Evaluating Logarithms with the Same Base

When the argument of a logarithm matches its base, such as log7(7), the value is 1 because any base raised to the power 1 equals itself. Recognizing this helps simplify expressions quickly without a calculator, as in log7(7/x) where log7(7) simplifies to 1.

Simplifying Algebraic Expressions Inside Logarithms

Understanding how to manipulate algebraic expressions inside logarithms is crucial. This includes factoring, dividing, or rewriting expressions to apply logarithmic properties effectively. For log7(7/x), recognizing the division inside the log allows the use of the quotient rule to separate terms.