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 $\text{<mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup> <mi>b</mi> <mi>x</mi> </msup>}$

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

Definition of the logarithmic function	For $\text{<mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>≠</mo><mn>1</mn><mo>,</mo>}$ $y={\log }_b\left(x\right)$ if and only if $\text{<msup> <mi>b</mi> <mi>y</mi> </msup> <mo>=</mo><mi>x</mi><mo>.</mo>}$
Definition of the common logarithm	For $\text{<mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo>}$ $y=\log \left(x\right)$ if and only if $\text{<msup> <mrow> <mn>10</mn> </mrow> <mi>y</mi> </msup> <mo>=</mo><mi>x</mi><mo>.</mo>}$
Definition of the natural logarithm	For $\text{<mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo>}$ $y=\ln \left(x\right)$ if and only if $\text{<msup> <mi>e</mi> <mi>y</mi> </msup> <mo>=</mo><mi>x</mi><mo>.</mo>}$

General Form for the Translation of the Parent Logarithmic Function $\text{<mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub> <mrow> <mi>log</mi> </mrow> <mi>b</mi> </msub> <mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow>}$

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

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 $\text{<mi>S</mi>}$ and $\text{<mi>T</mi>}$ and any positive real number $\text{<mi>b</mi><mo>,</mo>}$ where $b^S=b^T$ if and only if $\text{<mi>S</mi><mo>=</mo><mi>T</mi><mo>.</mo>}$
Definition of a logarithm	For any algebraic expression S and positive real numbers $\text{<mi>b</mi><mo> </mo>}$ and $\text{<mi>c</mi><mo>,</mo>}$ where $\text{<mi>b</mi><mo>≠</mo><mn>1</mn><mo>,</mo>}$ ${\log }_b(S)=c$ if and only if $\text{<msup> <mi>b</mi> <mi>c</mi> </msup> <mo>=</mo><mi>S</mi><mo>.</mo>}$
One-to-one property for logarithmic functions	For any algebraic expressions S and T and any positive real number $\text{<mi>b</mi><mo>,</mo>}$ where $\text{<mi>b</mi><mo>≠</mo><mn>1</mn><mo>,</mo>}$ ${\log }_{b}S={\log }_{b}T$ if and only if $\text{<mi>S</mi><mo>=</mo><mi>T</mi><mo>.</mo>}$

Half-life formula	If $\text{<mi>A</mi><mo>=</mo><msub> <mi>A</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mi>k</mi><mi>t</mi> </mrow> </msup> <mo>,</mo>}$ $k<0,$ the half-life is $\text{<mi>t</mi><mo>=</mo><mo>−</mo><mfrac> <mrow> <mi>ln</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo> </mrow> <mi>k</mi> </mfrac> <mo>.</mo>}$
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 $\text{<mi>A</mi><mo>=</mo><msub> <mi>A</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mi>k</mi><mi>t</mi> </mrow> </msup> <mo>,</mo>}$ $k>0,$ the doubling time is $\text{<mi>t</mi><mo>=</mo><mfrac> <mrow> <mi>ln</mi><mn>2</mn> </mrow> <mi>k</mi> </mfrac>}$
Newton’s Law of Cooling	$T(t)=A{e}^{kt}+T_s,$ where $\text{<msub> <mi>T</mi> <mi>s</mi> </msub>}$ is the ambient temperature, $\text{<mi>A</mi><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><msub> <mi>T</mi> <mi>s</mi> </msub> <mo>,</mo>}$ and $\text{<mi>k</mi>}$ is the continuous rate of cooling.