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 3: .
Example 6: Rewrite the term as the sum and difference of fully simplified terms.
Solution:
can be written
can be written
Is this true always? No, it is only true for certain values of x, y, w, and z.. Remember that you cannot take the log of any number less than or equal to 0. In the initial expression, we need to find values of x, y, w, and w so that . There are many combinations. In one case the y and the w could be negative. In the final expression, we need to find values of x, y, w, and z so that
can be calculated. In this case x, y, w, and z must all be greater than 0. You use the domain that is common to both expressions. Therefore, x, y, w, and z must all be greater than 0.
How could you check your answer? The initial expression and each subsequent expression are equivalent. This means that if we pick any positive numbers for x, y, w, and z and substitute them in each of the expressions, the values will be the same. Let's do it for x = 2, y = 3, w = 4, and z = 5.
Original Expression: The value of when we substitute the values for x, y, w, and z is
Step 1 Expression: The value of when we substitute the values for x, y, w, and z is
Step 2 Expression: The value of when we substitute the values for x, y, w, and z is
Step 3 Expression: The value of
when we substitute the values for x, y, w, and z is
Final Expression: The value of
when we substitute the values for x, y, w, and z is
We have illustrated our answer.
If you would like to review another example, click on Example.
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Author: Nancy Marcus