Transcript Document

Properties
of
Logarithms
Since logs and exponentials of the same base are
inverse functions of each other they “undo” each other.
f x   a
Remember that:
This means that:
inverses
“undo” each
each other
2
log 2 5
x
f
1
x  loga x
f  f 1  x and f 1  f  x
ff
1
a
loga x
f 1  f  loga a x  x
7
=5
x
log3 3
=7
Properties of
Logarithms
CONDENSED
1.
=
EXPANDED
loga MN  = loga M  loga N
2.
M
log a
N
= loga M  loga N
3.
loga M r
= r loga M
(these properties are based on rules of
exponents since logs = exponents)
Using the log properties, write the expression as a
sum and/or difference of logs (expand).
 ab
log6 
3 2
 c
4
 4
ab 

 log6  2 
 3 
 c 




When working with logs, re-write any radicals as rational
exponents.
using the second property:
log a
M
 log a M  log a N
N
using the first property:
loga MN  loga M  loga N
log6 a  log6 b 4  log6 c
using the third property:
loga M r  r loga M
log6 ab4  log6 c
2
3
2
3
2
log 6 a  4 log 6 b  log 6 c
3
Using the log properties, write the expression as a
single logarithm (condense).
1
2 log 3 x  log 3 y
2
using the third property:
loga M r  r loga M
log3 x 2  log3 y
this direction
using the second property:
log a
M
 log a M  log a N
N
this direction
log3
x
y
2
1
2
1
2
More Properties of Logarithms
This one says if you have an equation, you can take
the log of both sides and the equality still holds.
If M  N , thenloga M  loga N
If loga M  loga N , thenM  N
This one says if you have an equation and each side
has a log of the same base, you know the "stuff" you
are taking the logs of are equal.
log2 8  3
(2 to the what is 8?)
log2 16  4
(2 to the what is 16?)
 3.32
log2 10 
(2 to the what is 10?)
Check by
putting 23.32 in
your calculator
(we rounded so
it won't be
exact)
There is an answer to this and it must
be more than 3 but less than 4, but
we can't do this one in our head.
Let's put it equal to x and we'll solve for x.
Change to
exponential form.
log2 10  x
use log property & take log of
both sides (we'll use common log)
If M  N , thenloga M  loga N
use 3rd log property
2  10
x
log 2  log10
x
loga M r  r loga M
solve for x by
dividing by log 2
use calculator to
approximate
x log 2  log10
log10
 3.32
x
log 2
If we generalize the process we just did
we come up with the:
Change-of-Base Formula
logb M
loga M 
logb a
log M

log a
ln M

ln a
The base you change to can “common”
log base 10
be any base so generally
we’ll want to change to a
LOG
base so we can use our
calculator. That would be
LN
either base 10 or base e.
“natural” log
base e
Example
for TI-83
Use the Change-of-Base Formula and a calculator to
approximate the logarithm. Round your answer to three
decimal places.
log3 16
Since 32 = 9 and 33 = 27, our answer of what exponent
to put on 3 to get it to equal 16 will be something
between 2 and 3.
ln 16
log 3 16 
ln 3
 2.524
put in calculator
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au