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 given expression \( \log_7 \left( \frac{7}{x} \right) \), which becomes \( \log_7 7 - \log_7 x \).
3. Recognize that \( \log_7 7 \) asks the question: "To what power must 7 be raised to get 7?" Since 7 to the power of 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 using logarithmic properties 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 log_7(7/x).

Logarithm of the Base

The logarithm of a number to its own base is always 1, i.e., log_b(b) = 1. This fact helps simplify expressions where the argument of the logarithm is the base itself, such as log_7(7), which equals 1. Recognizing this can simplify parts of the expression without a calculator.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves using known values and properties to simplify expressions. For example, recognizing that log_7(7) = 1 and applying the quotient rule can reduce the expression to a simpler form. This skill is important for exact answers in algebraic contexts.