Exponential functions form the basis for understanding mathematical phenomena such as growth and decay. These functions, defined by a constant base raised to a variable exponent, are crucial in various fields, including finance and biology. The text delves into the core rules of exponents, contrasting exponential growth with decay, and provides practical examples of these concepts in action. Mastery of these rules is essential for simplifying complex expressions and predicting trends.
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Exponential functions are mathematical expressions in the form \( b^x \), where \( b \) is the base and \( x \) is the exponent
Product Rule
The product rule states that when multiplying expressions with the same base, the exponents should be added
Quotient Rule
The quotient rule states that when dividing expressions with the same base, the exponents should be subtracted
Power Rule
The power rule involves raising an exponentiated expression to another power, resulting in the multiplication of the exponents
Exponential functions are used to model phenomena such as growth and decay in fields like finance and biology
Exponential growth and decay are mathematical models that describe how quantities evolve over time
Exponential Growth Model
The exponential growth model is represented by \( f(x) = a \cdot b^{x} \) with \( a > 0 \) and \( b > 1 \)
Exponential Decay Model
The exponential decay model is represented by \( f(x) = a \cdot b^{x} \), where \( a > 0 \) and \( 0 < b < 1 \)
Exponential growth and decay models are used to understand dynamic processes in nature and technology, such as population growth and radioactive decay
The exponential product rule is a fundamental concept for simplifying expressions with exponents
The exponential product rule is used to simplify expressions with the same base by adding their exponents
The exponential product rule can only be applied when the bases are identical and cannot be used for expressions with different bases