Applications of 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 the base (a positive real number), and 'x' is the exponent. It describes growth or decay processes.

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What is the formula for compound interest?

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

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What does 'compounded continuously' mean?

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Compounding continuously means that interest is calculated and added to the principal at every moment, rather than at discrete intervals. The formula used is A = Pe^(rt), where e is the base of the natural logarithm.

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How do you calculate the future value of an investment?

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The future value of an investment can be calculated using the formula A = P(1 + r/n)^(nt) for compound interest or A = Pe^(rt) for continuous compounding.

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What is the difference between exponential growth and exponential decay?

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Exponential growth occurs when a quantity increases at a rate proportional to its current value, while exponential decay occurs when a quantity decreases at a rate proportional to its current value.

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What is the natural logarithm?

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The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is used to solve equations involving exponential growth and decay.

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How do you solve for time in continuous compounding?

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To solve for time in continuous compounding, use the formula t = (ln(A/P)) / r, where A is the amount of money accumulated, P is the principal amount, and r is the annual interest rate.

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What is the rule of 70?

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The rule of 70 is a way to estimate the number of years required to double the value of an investment at a fixed annual rate of return. It states that you can divide 70 by the annual interest rate (in percentage) to get the approximate number of years.

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What is the formula for calculating interest earned on an investment?

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The interest earned can be calculated using the formula Interest = A - P, where A is the total amount after interest and P is the principal amount.

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What is the significance of the base 'e' in exponential functions?

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The base 'e' is significant in exponential functions because it represents continuous growth or decay. It is the limit of (1 + 1/n)^n as n approaches infinity.