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: tex2html_wrap_inline90

Example 1: Suppose that a base is 6 and exponents are 3 and 5. We could solve the exponential problem tex2html_wrap_inline92 by calculating tex2html_wrap_inline94 and tex2html_wrap_inline96 and multiplying the results. tex2html_wrap_inline98 and tex2html_wrap_inline100 and their product is tex2html_wrap_inline102 . You could also solve the problem by first combing the exponents tex2html_wrap_inline104 .
The same is true of logarithms. Suppose you wanted to calculate tex2html_wrap_inline106 . 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 tex2html_wrap_inline112 .

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 tex2html_wrap_inline112 can be written tex2html_wrap_inline116 . By Rule 1 we can also write tex2html_wrap_inline112 as


To write this problem in turns of exponents, note that the base is 10 and the exponents are 1.4771213 and 0.69897. tex2html_wrap_inline122 , rounded to 30. tex2html_wrap_inline124 , 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 tex2html_wrap_inline128 .


Problem 2: Calculate tex2html_wrap_inline130 .


Problem 3: Calculate tex2html_wrap_inline132 .


Problem 4: Calculate tex2html_wrap_inline134 .


Problem 5: Simplify tex2html_wrap_inline136 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

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