Download Notes Logarithmic Functions

Notes Logarithmic Functions
Because Exponential Functions are one-to-one, they will have an inverse. The inverse to an exponential function is
a logarithmic function.
Definition of logs:
y  log a x if and only if a y  x . The base stays the same for both but we SWAP the x and the y.
NOTE:

If you do not have a base, for example,

We do not write
log x , you base is 10.
log e x , log e is written as ln
Properties of Logs:

log a a =_________________

ln e =_________________

log a 1 =_________________

ln1 =_________________

log a a r =_________________

ln er =_________________

a loga r =_________________

eln r =_________________
Rewrite the following in logarithmic form:
Ex1:
Ex3:
43  64
Ex2:
80  1
82
Ex4:
7 y  300
Ex6:
ln e  1
Ex8:
log5 125  y
3
Rewrite the following in exponential form:
Ex5:
log9 3  1
Ex7:
log10  1
2
Graph a logarithmic function
f 1 ( x)  2 x
x
Ex9: f ( x)  log 2 x
x
f(x)
Domain:
Range:
x-int:
y-int:
Vertical
Asymptote:
In general, if we have the function f ( x)  log a x it will have:

Domain:  0,  

Range: All real numbers  ,  

y-int: NONE

x-int:  0,1

Vertical Asymptote: y  0

All logarithmic functions (in “basic” form) have 2 anchor points: 1, 0  and  a,1
We can graph exponential functions by transformations. We will transform the anchor points and find
the new vertical asymptote. (The vertical asymptote is determined by the horizontal shift).
Obtain the graphs of the following functions by translating, reflecting, and stretching. Give the domain,
range, and equations of any asymptotes of the function.
Ex10: f ( x)  log x  2
Ex11: f ( x)  4  log 2 ( x  1)
V.A.: _____________________
V.A.: _____________________
Domain: __________________
Domain: __________________
Range:____________________
Range:____________________
Ex12: f ( x)  ln( x  2)  1
Ex13: f ( x)  2  ln( x  1)
V.A.: _____________________
V.A.: _____________________
Domain: __________________
Domain: __________________
Range:____________________
Range:____________________