Properties of Exponents and Logarithms Learning Activity
I. Evaluate each logarithm using the change of base formula: log b u  log( u)  log( b) or log b u  ln( u)  ln( b)
Round to the nearest thousandths, when necessary.
1. log 2 512
2. log 3 50
3. log 7 184
4. log 5 12,000
II. By the definition of a logarithm, it follows that the logarithmic function g ( x)  log b x is the inverse of the
exponential function f ( x)  b x . This means that exponential and logarithmic functions “undo” each other, so
it follows that:
A. g ( f ( x))  log b b x  x
B. f ( g ( x))  b logb x  x
C. Simplify the following using inverse rules:
5.
5log x
5
9. log 100 x
9log x
6. log 2 2 x
7. log 4 16 x
8.
10. log 3 81 x
11. log 2 16 x
12.
9
7 log x
7
III. Now, we’ll review the Properties of Exponents. Being familiar with these will enable you to develop and
understand the properties of logarithms we will explore next.
A. Product Property: a m * a n  a m n
B. Quotient Property:
am
 a mn
an
 
C. Power Property: a m
D.
n
 a mn
a0  1
E. a n 
1
an
F. Use the properties of exponents to simplify the following expressions:
13.
x x
2

1
2

17.  x 


5
3
 
14. x
x5
15. 2
x
2 3
1
2
18. x  x
1
2
19.
x3
16. 5
x
2
1
x3
x4
x
1
4
20.
2
x3
IV. Because of the relationship between exponents and logarithms, the Properties of Logarithms follow:
A. Product Property: log b uv  log b u  log b v
B. Quotient Property: log b
u
 log b u  log b v
v
C. Power Property: log b u  n log b u
n
D. When Ms. Ginn was in school, she was not allowed to use a calculator. She and other
students had to use a table of logarithms for prime numbers and the properties of logarithms in order
to calculate the value of any logarithmic expression. With the following, you, too, should be able to
apply the properties of logarithms in order to evaluate other logarithmic expressions:
log 2  .3010, log 3  .4771, log 5  0.6990, and log 7  0.8451
Evaluate the following logarithms to four decimal places without using a calculator.
21.
log 6
22.
log 8
23.
log 75
24.
log 49
E. Now, use the properties of logarithms to expand the following expressions:
25. log 2 3 x
25
x4
2 3
27. log 5 7 x y
26. log 3
28. ln 3 x
29. log 4
3
x3 y
F. And finally, use the properties of logarithms to condense the following expressions:
30. log 3 2  5 log 3 x
31. log 5 x  3 log 5 y
32. log 2 8  log 2 12  3 log 2 x
33. ln 2 
1
ln x
3
34. log 3 2  5 log 3 x  log 3 6  2 log 3 x