Virtual Day 12-9 Test Review-Logs and Exponentials

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What is exponential growth?

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Exponential growth occurs when the increase in a quantity is proportional to its current value, leading to growth at an increasing rate. It can be modeled by the equation f(t) = a * e^(kt), where 'a' is the initial amount, 'k' is the growth rate, and 't' is time.

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What is the formula for continuous compound interest?

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The formula for continuous compound interest is A = Pe^(rt), where A is the amount of money accumulated after time t, P is the principal amount (the initial investment), r is the annual interest rate (decimal), and t is the time in years.

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How do you find the percent increase in a population modeled by an exponential function?

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To find the percent increase, use the formula: Percent Increase = (e^k - 1) * 100%, where k is the growth rate in the exponential function f(t) = a * e^(kt).

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What is a logarithm?

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A logarithm is the inverse operation to exponentiation, indicating the power to which a base must be raised to obtain a given number. For example, log_b(a) = c means b^c = a.

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What is the change of base formula for logarithms?

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The change of base formula states that log_b(a) = log_k(a) / log_k(b) for any positive base k, allowing you to convert logarithms to a different base.

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What is the domain of the function f(x) = log(x + 1)?

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The domain of f(x) = log(x + 1) is x > -1, since the argument of the logarithm must be positive.

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What is the range of the function f(x) = log(x + 1)?

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The range of f(x) = log(x + 1) is (-∞, ∞), as logarithmic functions can take any real value.

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What does it mean for a function to be one-to-one?

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A function is one-to-one if it never takes the same value twice, meaning for every y in the range, there is exactly one x in the domain such that f(x) = y.

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How do you solve a logarithmic equation?

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To solve a logarithmic equation, you can use properties of logarithms to combine or separate logs, then exponentiate to eliminate the logarithm and solve for the variable.

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What is the formula for the sum of logarithms?

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The sum of logarithms can be expressed as log_b(m) + log_b(n) = log_b(m * n), where b is the base.