Final answer:
The question revolves around high school level mathematics, specifically the concepts of logarithms and their properties, such as calculation of order of magnitude and relationships with exponential functions.
Step-by-step explanation:
The subject of the question pertains to the understanding and application of logarithms, which are integral to high school level mathematics.
Given different mathematical expressions involving logarithms, we can deduce certain properties such as the order of magnitude of a number. For instance, the order of magnitude of 450 is 10³ because log10450 ≈ 2.653, which rounds to 3. This means that 450 shares the same order of magnitude with 800, since both round to 10³. Comparatively, the order of magnitude of 250 is 10², based on its logarithm log10250 ≈ 2.397 rounding to 2.
Logarithms feature properties that aid in simplifying mathematical calculations. For example, log(a) = log(a) - log(b) applies to any base, meaning log10 and ln function similarly. Utilizing this property, understanding the relationship between exponentials and logarithms is crucial as they are inverse functions. Hence, ln(ex) = x and eln(x) = x.
Additionally, any base b raised to a power n can be represented as bn = en ln(b) = 10n.log10(b), which is useful for converting between different base representations. This concept is applicable when dealing with growth, as it allows for the computing of attributes like doubling time (t2) through logarithmic equations such as ln(1 + p) = ln 2/t2.