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. Identify the logarithmic expression given: \(\log_{7}(7x)\).
2. Recall the logarithmic property that states \(\log_b(mn) = \log_b(m) + \log_b(n)\), which allows us to expand the log of a product into a sum of logs.
3. Apply this property to the expression: \(\log_{7}(7x) = \log_{7}(7) + \log_{7}(x)\).
4. Evaluate \(\log_{7}(7)\) using the fact that \(\log_b(b) = 1\) for any base \(b\), so \(\log_{7}(7) = 1\).
5. Write the fully expanded form as \$1 + \log_{7}(x)$, which is the simplified expanded expression.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules that allow the expansion or simplification of logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N) helps break down complex expressions into simpler parts.

Logarithm of a Base Raised to a Power

The logarithm of a base raised to the same base simplifies to the exponent, i.e., log_b(b) = 1. This property is useful for evaluating expressions like log_7(7), which equals 1, simplifying the overall expression.

Evaluating Logarithmic Expressions Without a Calculator

Some logarithmic expressions can be evaluated exactly by recognizing patterns or using properties, such as when the argument is a power of the base. This avoids approximation and provides exact values, enhancing understanding of logarithmic behavior.