Exponential and Logarithmic Functions

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a * b^x, where a is a constant, b is a positive real number, and x is the exponent. It models growth or decay processes.

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What is the base of an exponential function?

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The base of an exponential function is the constant b in the function f(x) = a * b^x. It determines the rate of growth (b > 1) or decay (0 < b < 1).

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What is the general form of an exponential decay function?

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The general form of an exponential decay function is f(x) = a * e^(-kx), where a is the initial amount, k is a positive constant, and e is Euler's number (approximately 2.718).

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What is the difference between exponential growth and decay?

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Exponential growth occurs when the base of the exponential function is greater than 1 (b > 1), while exponential decay occurs when the base is between 0 and 1 (0 < b < 1).

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What is a logarithmic function?

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A logarithmic function is the inverse of an exponential function, expressed as f(x) = log_b(x), where b is the base. It answers the question: to what exponent must the base b be raised to produce x?

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What is the natural logarithm?

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The natural logarithm is a logarithm with base e (approximately 2.718), denoted as ln(x). It is commonly used in calculus and exponential growth/decay problems.

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What is the horizontal asymptote of an exponential function?

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The horizontal asymptote of an exponential function f(x) = a * b^x is the line y = 0, which the graph approaches as x approaches negative infinity.

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How do you find the equation of a horizontal asymptote?

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To find the equation of a horizontal asymptote for a function, identify the value that the function approaches as x approaches infinity or negative infinity.

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What is the compound interest formula?

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The compound interest formula is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

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How do you model exponential growth in real-life situations?

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Exponential growth can be modeled using the function f(t) = A_0 * e^(rt), where A_0 is the initial amount, r is the growth rate, and t is time.