### Key Equations

definition of the exponential function | $f(x)={b}^{x}\text{,where}b0,b\ne 1$ |

definition of exponential growth | $f(x)=a{b}^{x},\phantom{\rule{0.5em}{0ex}}\text{where}a0,b0,b\ne 1$ |

compound interest formula | $\begin{array}{l}A(t)=P{\left(1+\frac{r}{n}\right)}^{nt},\phantom{\rule{0.5em}{0ex}}\text{where}\hfill \\ A(t)\phantom{\rule{0.5em}{0ex}}\text{istheaccountvalueattime}t\hfill \\ t\phantom{\rule{0.5em}{0ex}}\text{isthenumberofyears}\hfill \\ P\phantom{\rule{0.5em}{0ex}}\text{istheinitialinvestment,oftencalledtheprincipal}\hfill \\ r\phantom{\rule{0.5em}{0ex}}\text{istheannualpercentagerate(APR),ornominalrate}\hfill \\ n\phantom{\rule{0.5em}{0ex}}\text{isthenumberofcompoundingperiodsinoneyear}\hfill \end{array}$ |

continuous growth formula | $A(t)=a{e}^{rt},\phantom{\rule{0.5em}{0ex}}\text{where}$ $t$ 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 $\text{}f(x)={b}^{x}$ | $f(x)=a{b}^{x+c}+d$ |

Definition of the logarithmic function | For $\text{}x0,b0,b\ne 1,$ $y={\mathrm{log}}_{b}\left(x\right)$ if and only if $\text{}{b}^{y}=x.$ |

Definition of the common logarithm | For $\text{}x>0,$ $y=\mathrm{log}\left(x\right)$ if and only if $\text{}{10}^{y}=x.$ |

Definition of the natural logarithm | For $\text{}x>0,$ $y=\mathrm{ln}\left(x\right)$ if and only if $\text{}{e}^{y}=x.$ |

General Form for the Translation of the Parent Logarithmic Function $\text{}f(x)={\mathrm{log}}_{b}\left(x\right)$ | $f(x)=a{\mathrm{log}}_{b}\left(x+c\right)+d$ |

The Product Rule for Logarithms | ${\mathrm{log}}_{b}(MN)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)$ |

The Quotient Rule for Logarithms | ${\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}M-{\mathrm{log}}_{b}N$ |

The Power Rule for Logarithms | ${\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M$ |

The Change-of-Base Formula | ${\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\phantom{\rule{0.5em}{0ex}}\text{}n0,n\ne 1,b\ne 1$ |

One-to-one property for exponential functions | For any algebraic expressions $\text{}S$ and $\text{}T$ and any positive real number $\text{}b,$ where ${b}^{S}={b}^{T}$ if and only if $\text{}S=T.$ |

Definition of a logarithm | For any algebraic expression S and positive real numbers $\text{}b$ and $\text{}c,$ where $\text{}b\ne 1,$ ${\mathrm{log}}_{b}(S)=c$ if and only if $\text{}{b}^{c}=S.$ |

One-to-one property for logarithmic functions | For any algebraic expressions S and T and any positive real number $\text{}b,$ where $\text{}b\ne 1,$ ${\mathrm{log}}_{b}S={\mathrm{log}}_{b}T$ if and only if $\text{}S=T.$ |

Half-life formula | If $\text{}A={A}_{0}{e}^{kt},$ $k<0,$ the half-life is $\text{}t=-\frac{\mathrm{ln}(2)}{k}.$ |

Carbon-14 dating | $t=\frac{\mathrm{ln}\left(\frac{A}{{A}_{0}}\right)}{-0.000121}.$
${A}_{0}$ 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 $\text{}A={A}_{0}{e}^{kt},$ $k>0,$ the doubling time is $\text{}t=\frac{\mathrm{ln}2}{k}$ |

Newton’s Law of Cooling | $T(t)=A{e}^{kt}+{T}_{s},$ where $\text{}{T}_{s}$ is the ambient temperature, $\text{}A=T(0)-{T}_{s},$ and $\text{}k$ is the continuous rate of cooling. |