Pre Cal Logs
The power rule for logarithms states that logbMp= plogbM
The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number
The quotient rule for logarithms states that logb (M/N)=LogsmallbM -logsmallb N
The logarithm of a quotient is the difference of the logarithms
n=number of levels number of objects on bottem level= 2^n
To find total number of objects 2^n+1 -1
xln9=ln20
x=ln20/ln9
log^4x-1=log^7
4x-1=7 because common base
The product rule for logarithms states that logb (MN)
=logsmallb M +logsmallb N. The logarithm of a product is the sum of the logarithms
logb(cx)
horizontal stretch if c>1 horizontal compression if 0<c<1
Logarithmic function with base e
is called the natural logarithmic function and is defined as lnx
the value (1+ 1/n)^n
is the natural base equal to 2.72
Irrational number e, aproxiomately 2.72, called the
natural base
x^4=15
not exponential because x input not on exponent
Inverse functions are
switched domain and range
if e^0.6x=6
then 0.6x=ln6
if logunder5(x+1)=3
then 5^3=x+1
if horizontal asymptote: y=0
vertical asymptote: x=0
clogbx
vertical stretch if c>1 vertical compression if 0<c<1
Logarithmic Functions
x >0 and b>0 and b cannot equal 1
y int of a log is the
x intercept of a exp
only has y intercept no x intercept
(0,1)
DomaIn and range
(0,infinite) and infinite and infinite
domain and range
(infinite) (0,infinite)
one to one function means passes horizontal line test
And has an inverse
Exponential Function
Characteristics
x intercept of logs (1,0)
Do not have a y intercept
X lies
Here too
remember toi
INTERCHANGE BEFORE SOLVIG
B>1
Increases to right
b
Is the base y=logbx
When they say verify that p= 3
Just plug in ANY COORDINATE FROM THE GRAPH into the formula
The change of base property for logarithms allows a logarithm with base b to be written in terms of a new base (a),
LogbM= LogM/logB
b^M=b^N
N=M
Approacches y axis but does not touch it
Y axis becomes vertical asymtote: x=0
can be transformed
and has horizontal asymptote
basically b is a constant other than one
b>0 and b is not one in both logs and exponential however x is greater then 0 in logs but in exponential is any real number
y=b^x
b>0 and not 1 and x has to be any real number
Gradient of curve at f(x)
becomes steeper
0<b<1
decreases to right
e is used to
describe growth and decay
The exponential function f with base b is defined by f (x)= b^x b>0 and b cannot equal 1
domain (-infinite, infinite) range: (0, infinite)
4^x=15
exponential because x input is on exponent
e, the natural base or number signifies
growth over time
