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Key Equations

Key Equations

definition of the exponential function f(x)= b x ,  where  b>0, b≠1 f(x)= b x ,  where  b>0, b≠1
definition of exponential growth f(x)=a b x ,where a>0, b>0, b≠1 f(x)=a b x ,where a>0, b>0, b≠1
compound interest formula A(t)=P ( 1+ r n ) nt  ,where A(t)is the account value at time t tis the number of years Pis the initial investment, often called the principal ris the annual percentage rate (APR), or nominal rate nis the number of compounding periods in one year A(t)=P ( 1+ r n ) nt  ,where A(t)is the account value at time t tis the number of years Pis the initial investment, often called the principal ris the annual percentage rate (APR), or nominal rate nis the number of compounding periods in one year
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  x>0,b>0,b≠1,  x>0,b>0,b≠1,
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 )  f(x)=a log b ( x+c )+d  f(x)=a log b ( x+c )+d
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 log b M= log n M log n b         n>0,n≠1,b≠1 log b M= log n M log n b         n>0,n≠1,b≠1
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 b  b  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.