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
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