Properties of Logarithms
One of the greatest pop bands of the late 80’s early 90’s was Ace
of Base. They created such hits as Cruel Summer featured in The
Karate Kid, a movie starting New Jersey native Ralph Machio,
who played Daniel. The movie was based on the decision of his
mother to move from New Jersey to California and it dealt with the
culture shock that the young man felt as he was transplanted.
Daniel was having a tough time in his new school and was
befriended by an older man who taught him to balance his life
through karate lessons.
I digress. We use the Ace of Base in math in order to evaluate a
logarithm on the calculator. We call the formula by its more
common name, the Change of Base.
log a x 
ln x
ln a
Try these:
log 5 12 
log 7 3 
ln 12

ln 5
1.54
ln 3

ln 7
0.565
The definition of a logarithmic function is:
For x  0, a  0, and a  1,
y  log a x if and only if a y  x
f ( x)  log a x is called the logarthmic function
Remember, there are different styles of logarithmic functions
Logarithmic Style log a x  y
Exponential Style a y  x
Gangum Style
PSY
We use these styles to formulate the rules that we use to evaluate
logarithms:
Properties of Logarithms
1. log a 1  0 because a 0  1
2. log a a  1 because a 1  a
3. log a a x  x
4. a loga x  x
5. If log a x  log a y, then x  y
Property 3 and 4 are inverse properties and property 5 is one-toone.
There is also a natural logarithmic function:
For x > 0,
y  ln x if and only if x  e y
The function, called the natural logarithmic function, is given by:
f ( x)  log e x  ln x
Of course, this function has its own properties, which are very
similar to the properties of the logarithmic function:
Properties of Natural Logarithms
1. ln 1  0 because e 0  1
2. ln e  1 because e 1  e
3. ln e x  x
4. e ln x  x
5. If ln x  ln y, then x  y
Once again, 3 and 4 represent inverse properties while 5 represents
the one-to-one property.
Now that we have discussed the properties that we will enhance
our understanding of logarithms, we have to figure out how to use
them. Or, “Does this answer make sense in the context of the
problem?” We need to determine how to establish the domain!
When we look at the graph of the natural log, we see that its
domain is (0, ) . That is a starting point in our discussion of the
domain.
Determine the domain for the following functions:
1. f ( x)  ln( x  2)
Set
x20
x  2
so the domain would be (2, )
2. f ( x)  ln( 3x  2)
3x  2  0
2
Set 3 x  2
so the domain is  ,  
3 
2
x
3
3. f ( x)  x 2
x2  0
Set
so the domain is (,0)  (0, )
x  0 or x  0
What happens if we define the domain?
ExampleStudents in a math class were given an exam and then retested
monthly with an equivalent exam. The average scores for the class
are given by the human memory model:
f (t )  78  17 log(t  1),0  t  12
a. What was the score on the original exam?
f (0)  78  17 log( 0  1)
f (0)  78  17 log(1)
f (0)  78  17(0)
f (0)  78
b. What was the average score after 3 months?
f (3)  78  17 log(3  1)
f (3)  78  17 log( 4)
f (3)  67.8
c. What was the average score after 11 months?
f (11)  78  17 log(11  1)
f (11)  78  17 log(12)
f (11)  59.7
Based on this data, what can you conclude about the students and
their reaction to this exam?