RULES OF LOGARITHMS - Rule 1
RULES OF LOGARITHMS - Rule 1
Let a be a positive number such that a does not equal 1, let
n be a real number, and let u and v be positive real numbers.
Logarithmic Rule 1: 
Example 1: Suppose that a base is 6 and exponents are 3 and 5. We could solve the exponential problem 
by calculating 
and
and multiplying the results. 
and 
and their product is 
. You
could also solve the problem by first combing the exponents
.
The same is true of logarithms. Suppose you wanted to calculate
. You could
calculate the answer by first multiplying 216 by 7776, changing the base of
6 to either 10 or e and calculating the results.
Or you could first combine the logarithms using Rule 1 and then change the bases.
rounded to 8.
Example 2: Calculate 
.
Solution: Since a base is not indicated, we know that the base is 10. We can calculate the logarithms directly by first multiplying the 30 and the 5. Note that can be
written 
. By Rule 1 we can also write as
To write this problem in turns of exponents, note that the base is 10 and
the exponents are 1.4771213 and 0.69897. 
,
rounded to 30. 
, rounded to 5. We could also
combine the bases
If you would like to review another example, click on
Example.
Work the following problems. If you would like to review the answers and solutions, click on answer.
Problem 1: Calculate 
.
Problem 2: Calculate 
.
Problem 3: Calculate 
.
Problem 4: Calculate 
.
Problem 5: Simplify 
and
write the answer in terms of a base 10. What assumptions must be made about
a, b, d, and d before you can work this problem?
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Author: Nancy Marcus