Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log N^-6

Verified step by step guidance

1. Recognize that the expression is \( \log N^{-6} \), which involves a logarithm of a power.
2. Use the logarithmic property that \( \log a^b = b \log a \) to rewrite the expression.
3. Apply this property to get \( \log N^{-6} = -6 \log N \).
4. Since the expression is now expanded, check if \( \log N \) can be simplified further or evaluated without a calculator (usually it cannot unless \(N\) is a known value).
5. Conclude that the fully expanded form of the expression is \( -6 \log N \).

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 power rule states that log(a^b) = b * log(a), which is essential for expanding expressions with exponents.

Negative Exponents in Logarithms

A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^{-b} = 1/a^b. When applied inside a logarithm, this affects the expression by introducing a negative sign outside the logarithm, as log(a^{-b}) = -b * log(a). Understanding this helps in correctly expanding and simplifying the expression.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing logarithms of numbers that are powers of the base or using known logarithmic values. Simplifying expressions using properties can reduce complex logs to simpler forms or constants, making it possible to evaluate them exactly without computational tools.