College Algebra

# Key Equations

College AlgebraKey Equations

### Key Equations

 definition of the exponential function definition of exponential growth compound interest formula continuous growth formula $A(t)=a e rt ,where A(t)=a e rt ,where$ $t t$ is the number of unit time periods of growth $a a$ is the starting amount (in the continuous compounding formula a is replaced with P, the principal) $e e$ is the mathematical constant, $e≈2.718282 e≈2.718282$
 General Form for the Translation of the Parent Function $f(x)= b x f(x)= b x$ $f(x)=a b x+c +d f(x)=a b x+c +d$
 Definition of the logarithmic function For $y= log b ( x ) y= log b ( x )$ if and only if $b y =x. b y =x.$ Definition of the common logarithm For $x>0, x>0,$ $y=log( x ) y=log( x )$ if and only if $10 y =x. 10 y =x.$ Definition of the natural logarithm For $x>0, x>0,$ $y=ln( x ) y=ln( x )$ if and only if $e y =x. e y =x.$
 General Form for the Translation of the Parent Logarithmic Function $f(x)= log b ( x ) f(x)= log b ( x )$
 The Product Rule for Logarithms $log b (MN)= log b ( M )+ log b ( N ) log b (MN)= log b ( M )+ log b ( N )$ The Quotient Rule for Logarithms $log b ( M N )= log b M− log b N log b ( M N )= log b M− log b N$ The Power Rule for Logarithms $log b ( M n )=n log b M log b ( M n )=n log b M$ The Change-of-Base Formula
 One-to-one property for exponential functions For any algebraic expressions $S S$ and $T T$ and any positive real number $b, b,$ where $b S = b T b S = b T$ if and only if $S=T. S=T.$ Definition of a logarithm For any algebraic expression S and positive real numbers and $c, c,$ where $b≠1, b≠1,$ $log b (S)=c log b (S)=c$ if and only if $b c =S. b c =S.$ One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number $b, b,$ where $b≠1, b≠1,$ $log b S= log b T log b S= log b T$ if and only if $S=T. S=T.$
 Half-life formula If $A= A 0 e kt , A= A 0 e kt ,$ $k<0, k<0,$ the half-life is $t=− ln(2) k . t=− ln(2) k .$ Carbon-14 dating $t= ln( A A 0 ) −0.000121 . t= ln( A A 0 ) −0.000121 .$ $A 0 A 0$ is the amount of carbon-14 when the plant or animal died $A A$ is the amount of carbon-14 remaining today $t t$ is the age of the fossil in years Doubling time formula If $A= A 0 e kt , A= A 0 e kt ,$ $k>0, k>0,$ the doubling time is $t= ln2 k t= ln2 k$ Newton’s Law of Cooling $T(t)=A e kt + T s , T(t)=A e kt + T s ,$ where $T s T s$ is the ambient temperature, $A=T(0)− T s , A=T(0)− T s ,$ and $k k$ is the continuous rate of cooling.
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