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 (7x)

Verified step by step guidance

1. Recall the logarithmic property that states \( \log_b (MN) = \log_b M + \log_b N \). This means the logarithm of a product can be expanded into the sum of the logarithms.
2. Apply this property to the expression \( \log_7 (7x) \), treating \(7x\) as the product of 7 and \(x\). So, \( \log_7 (7x) = \log_7 7 + \log_7 x \).
3. Recognize that \( \log_7 7 \) asks the question: 'To what power must 7 be raised to get 7?' Since \(7^1 = 7\), \( \log_7 7 = 1 \).
4. Substitute this value back into the expression to get \( 1 + \log_7 x \).
5. The expression is now fully expanded using logarithmic properties: \( \log_7 (7x) = 1 + \log_7 x \).

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 product rule states that log_b(MN) = log_b(M) + log_b(N), which is essential for expanding expressions like log7(7x).

Evaluating Logarithms with Known Bases

When the argument of a logarithm matches its base raised to a power, the logarithm can be evaluated directly. For instance, log7(7) equals 1 because 7^1 = 7. Recognizing such cases helps simplify expressions without a calculator.

Logarithmic Expression Expansion

Expanding logarithmic expressions involves breaking down complex arguments into sums or differences of simpler logarithms using properties. This process makes it easier to simplify or evaluate the expression, especially when combined with known values or further algebraic manipulation.