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.log(1000x)

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). Understanding these properties is essential for expanding logarithmic expressions effectively.

When dealing with logarithms, the logarithm of a power allows us to simplify expressions involving exponents. For example, log(a^b) can be rewritten as b*log(a). This concept is particularly useful when evaluating logarithmic expressions that include variables raised to powers, as it helps break down complex terms into simpler components.

Evaluating logarithmic expressions involves finding the value of the logarithm for specific inputs. For instance, log(1000) can be evaluated by recognizing that 1000 is 10^3, leading to log(1000) = 3. This skill is important for simplifying logarithmic expressions and understanding their numerical values without relying on calculators.