Exponential 7.1 & 7.2 Review

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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 variable. The base b determines the growth or decay rate.

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What does it mean for a function to be increasing or decreasing?

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A function is increasing if its output values rise as the input values increase. Conversely, it is decreasing if its output values fall as the input values increase.

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How do you solve exponential inequalities?

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To solve exponential inequalities, rewrite the inequality in terms of the same base, if possible, and then solve for the variable. Consider the direction of the inequality when multiplying or dividing by negative numbers.

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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, 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 is the decay factor in an exponential decay model?

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The decay factor is the base of the exponential function that represents the rate of decay. For example, in f(x) = a(1 - r)^x, the decay factor is (1 - r), where r is the decay rate.

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How do you convert between different bases in exponential equations?

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To convert between different bases, express both sides of the equation with the same base, if possible, and then set the exponents equal to each other.

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What is the relationship between exponential growth and decay?

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Exponential growth occurs when the base of the exponential function is greater than 1, leading to an increase over time. Exponential decay occurs when the base is between 0 and 1, leading to a decrease over time.

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How do you find the solution set of an exponential inequality?

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To find the solution set, isolate the variable in the inequality, convert to the same base if possible, and determine the range of values that satisfy the inequality.

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What is the significance of the number e in exponential functions?

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The number e (approximately 2.718) is the base of natural logarithms and is significant in continuous growth or decay processes, such as population growth and radioactive decay.

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How do you solve equations with the same base?

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If two exponential expressions have the same base, set their exponents equal to each other and solve for the variable.