Key Equations

Key Equations

definition of the exponential function	$f(x)=b^x\text{, where }b>0, b\ne 1$
definition of exponential growth	$f(x)=a{b}^x,\text{where }a>0,b>0,b\ne 1$
compound interest formula	$\begin{array}{l}A(t)=P{\left(1+\frac{r}{n}\right)}^{nt} ,\text{where}\hfill \\ A(t)\text{is the account value at time }t\hfill \\ t\text{is the number of years}\hfill \\ P\text{is the initial investment, often called the principal}\hfill \\ r\text{is the annual percentage rate (APR), or nominal rate}\hfill \\ n\text{is the number of compounding periods in one year}\hfill \end{array}$
continuous growth formula	$A(t)=a{e}^{rt},\text{where}$ $t$ 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 $x>0,b>0,b\ne 1,$ $y={\log }_b\left(x\right)$ if and only if $b^y=x.$
Definition of the common logarithm	For $x>0,$ $y=\log \left(x\right)$ if and only if ${10}^y=x.$
Definition of the natural logarithm	For $x>0,$ $y=\ln \left(x\right)$ if and only if $e^y=x.$

General Form for the Translation of the Parent Logarithmic Function $f(x)={\log }_b\left(x\right)$

$f(x)=a{\log }_b\left(x+c\right)+d$

The Product Rule for Logarithms	${\log }_b(MN)={\log }_b\left(M\right)+{\log }_b\left(N\right)$
The Quotient Rule for Logarithms	${\log }_b\left(\frac{M}{N}\right)={\log }_{b}M-{\log }_{b}N$
The Power Rule for Logarithms	${\log }_b\left(M^n\right)=n{\log }_{b}M$
The Change-of-Base Formula	${\log }_{b}M\text{=}\frac{{\log }_{n}M}{{\log }_{n}b}n>0,n\ne 1,b\ne 1$

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 $b\ne 1,$ ${\log }_b(S)=c$ 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 $b\ne 1,$ ${\log }_{b}S={\log }_{b}T$ if and only if $S=T.$

Half-life formula	If $A=A_0{e}^{kt},$ $k<0,$ the half-life is $t=-\frac{\ln (2)}{k}.$
Carbon-14 dating	$t=\frac{\ln \left(\frac{A}{A_0}\right)}{-0.000121}.$ $A_0$ 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 $A=A_0{e}^{kt},$ $k>0,$ the doubling time is $t=\frac{\ln 2}{k}$
Newton’s Law of Cooling	$T(t)=A{e}^{kt}+T_s,$ where $T_s$ is the ambient temperature, $A=T(0)-T_s,$ and $k$ is the continuous rate of cooling.