4.10.2: Key Equations

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Key Equations

definition of the exponential function	$f (x) = b^{x}, where b > 0, b \neq 1$
definition of exponential growth	$f (x) = a b^{x}, where a > 0, b > 0, b \neq 1$
compound interest formula	$\begin{array}{l} A (t) = P {(1 + \frac{r}{n})}^{n t}, where \\ A (t) is the account value at time t \\ t is the number of years \\ P is the initial investment, often called the principal \\ r is the annual percentage rate (APR), or nominal rate \\ n is the number of compounding periods in one year \end{array}$
continuous growth formula	$A (t) = a e^{r t}, where$ $t$ is the number of unit time periods of growth $a$ is the starting amount (in the continuous compounding formula a is replaced with P, the principal) $e$ is the mathematical constant, $e \approx 2.718282$

General Form for the Translation of the Parent Function

f (x) = b^{x}

f (x) = a b^{x + c} + d

Definition of the logarithmic function	For $x > 0, b > 0, b \neq 1,$ $y = \log_{b} (x)$ if and only if $b^{y} = x .$
Definition of the common logarithm	For $x > 0,$ $y = \log (x)$ if and only if $10^{y} = x .$
Definition of the natural logarithm	For $x > 0,$ $y = \ln (x)$ if and only if $e^{y} = x .$

General Form for the Translation of the Parent Logarithmic Function

f (x) = \log_{b} (x)

f (x) = a \log_{b} (x + c) + d

The Product Rule for Logarithms	$\log_{b} (M N) = \log_{b} (M) + \log_{b} (N)$
The Quotient Rule for Logarithms	$\log_{b} (\frac{M}{N}) = \log_{b} M - \log_{b} N$
The Power Rule for Logarithms	$\log_{b} (M^{n}) = n \log_{b} M$
The Change-of-Base Formula	$\log_{b} M = \frac{\log_{n} M}{\log_{n} b} n > 0, n \neq 1, b \neq 1$

One-to-one property for exponential functions	For any algebraic expressions $S$ and $T$ and any positive real number $b,$ where $b^{S} = b^{T}$ if and only if $S = T .$
Definition of a logarithm	For any algebraic expression S and positive real numbers $b$ and $c,$ where $b \neq 1,$ $\log_{b} (S) = c$ if and only if $b^{c} = S .$
One-to-one property for logarithmic functions	For any algebraic expressions S and T and any positive real number $b,$ where $b \neq 1,$ $\log_{b} S = \log_{b} T$ if and only if $S = T .$

Half-life formula	If $A = A_{0} e^{k t},$ $k < 0,$ the half-life is $t = - \frac{\ln (2)}{k} .$
Carbon-14 dating	$t = \frac{\ln (\frac{A}{A_{0}})}{- 0.000121} .$ $A_{0}$ is the amount of carbon-14 when the plant or animal died $A$ is the amount of carbon-14 remaining today $t$ is the age of the fossil in years
Doubling time formula	If $A = A_{0} e^{k t},$ $k > 0,$ the doubling time is $t = \frac{\ln 2}{k}$
Newton’s Law of Cooling	$T (t) = A e^{k t} + T_{s},$ where $T_{s}$ is the ambient temperature, $A = T (0) - T_{s},$ and $k$ is the continuous rate of cooling.

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Key Equations