Unit 2: Exponential and Logarithmic Functions

19 terms. Say what each one means, then open the card to check.

• Arithmetic sequence

  A list of numbers in which each term is the previous term plus the same fixed amount, so the terms follow a linear pattern.

  Example. The seats per row in a theater run 20, 23, 26, 29, so each row adds 3 seats.

• Common difference

  The fixed amount added to get from one term of an arithmetic sequence to the next. It plays the role of slope.

  Example. In 40, 34, 28, 22 the common difference is -6, since each term is 6 less than the one before.

• Geometric sequence

  A list of numbers in which each term is the previous term times the same nonzero number, so the terms follow an exponential pattern.

  Example. A chain message sent to 2 people, then 6, then 18, then 54 multiplies by 3 at each round.

• Common ratio

  The fixed number each term of a geometric sequence is multiplied by to get the next term. Divide any term by the one before it to find it.

  Example. In 80, 20, 5, 1.25 the common ratio is 0.25, because 20 divided by 80 is 0.25.

• Exponential function

  A function of the form $f(x) = ab^x$ with $a \neq 0$, $b > 0$ and $b \neq 1$. The input sits in the exponent.

  Example. The function $f(x) = 250(1.04)^x$ starts at 250 and is multiplied by 1.04 each time x goes up by 1.

• Initial value

  The output of an exponential model when the input is 0. In $f(x) = ab^x$ it is the constant a.

  Example. If a pond holds $P(t) = 90(1.2)^t$ frogs t months after a survey, 90 frogs were counted at the survey.

• Growth factor

  The base b in $f(x) = ab^x$: the number the output is multiplied by when the input rises by 1. If b is above 1 the model grows; if b is between 0 and 1 it decays.

  Example. A factor of 0.9 per hour means the amount left each hour is 90 percent of the hour before.

• Percent rate of change

  The constant percent by which an exponential model changes per unit of input. The growth factor equals 1 plus this rate written as a decimal.

  Example. A growth factor of 1.06 is a 6 percent increase per unit, and a factor of 0.75 is a 25 percent decrease per unit.

• Half-life

  The time it takes a quantity that decays exponentially to fall to half of its value. It is the same no matter where you start.

  Example. A substance with a 4-hour half-life drops from 60 mg to 30 mg in 4 hours, then to 15 mg after 4 more.

• Composite function

  A function made by feeding the output of one function into another. $f(g(x))$ applies g first, then f.

  Example. If $g(x) = x + 2$ and $f(x) = 3x$, then $f(g(1)) = f(3) = 9$, while $g(f(1)) = g(3) = 5$.

• Inverse function

  The function that undoes f, written $f^{-1}$. If $f(a) = b$ then $f^{-1}(b) = a$, and its graph is the graph of f reflected across $y = x$.

  Example. If $f(x) = 2x + 6$, then $f^{-1}(x) = \frac{x - 6}{2}$, so f sends 1 to 8 and the inverse sends 8 back to 1.

• One-to-one function

  A function in which no two different inputs share an output. Only a one-to-one function, or one restricted to such a domain, has an inverse function.

  Example. The function $y = x^2$ is not one-to-one, since 3 and -3 both give 9, but it is one-to-one for $x \geq 0$.

• Logarithm

  The exponent you put on a base to get a number: $\log_b c = a$ means $b^a = c$. The logarithm is the inverse of the exponential function with the same base.

  Example. Since $2^5 = 32$, $\log_2 32 = 5$. The log answers the question: 2 to what power is 32.

• Natural logarithm

  The logarithm with base e, about 2.718, written $\ln x$. It is the inverse of $e^x$.

  Example. Since $e^0 = 1$, $\ln 1 = 0$, and $\ln(e^{3}) = 3$.

• Product property of logarithms

  For positive m and n, $\log_b(mn) = \log_b m + \log_b n$: multiplying inside a log becomes adding outside it.

  Example. Since $\log_{10} 20 = \log_{10} 2 + \log_{10} 10$, it equals about $0.301 + 1 = 1.301$.

• Change of base

  The rule $\log_b x = \frac{\ln x}{\ln b}$ for any positive base other than 1. It lets you compute any logarithm with the natural log.

  Example. To find $\log_3 50$, compute $\frac{\ln 50}{\ln 3}$, which is about 3.56.

• Extraneous solution

  A value produced by the algebra of solving an equation that does not satisfy the original equation, often because a log of it would be undefined.

  Example. Solving $\log_3 x + \log_3(x + 8) = 2$ gives 1 and -9, but -9 is rejected since $\log_3(-9)$ is undefined.

• Semi-log plot

  A graph whose vertical axis is on a logarithmic scale, or that plots the log of y against x. Exponential data look like a straight line on it.

  Example. Plotting $\log_{10} y$ against x for $y = 5(2)^x$ gives a line with slope $\log_{10} 2$, about 0.301.

• Linearization

  Rewriting data or a model so it becomes linear, such as taking the log of y. A linear fit can then be turned back into an exponential model.

  Example. If $\log_{10} y = 1 + 0.2x$, then $y = 10(10^{0.2})^x$, an exponential model with factor about 1.58.