Unformatted text preview: rocess, which we will
practice later, that is most useful in Algebra, the utility of expanding logarithms becomes apparent
Example 6.2.1. Expand the following using the properties of logarithms and simplify. Assume
when necessary that all quantities represent positive real numbers.
x 1. log2 4. log 3 100x2
yz 5 2. log0.1 10x2
3. ln 3
ex 2 5. log117 x2 − 4 Solution.
1. To expand log2 8
x , we use the Quotient Rule identifying u = 8 and w = x and simplify. 6.2 Properties of Logarithms 8
x log2 349 = log2 (8) − log2 (x) Quotient Rule
= 3 − log2 (x) Since 23 = 8 = − log2 (x) + 3
2. In the expression log0.1 10x2 , we have a power (the x2 ) and a product. In order to use the
Product Rule, the entire quantity inside the logarithm must be raised to the same exponent.
Since the exponent 2 applies only to the x, we ﬁrst apply the Product Rule with u = 10 and
w = x2 . Once we get the x2 by itself inside the log, we may apply the Power Rule with u = x
and w = 2 and simplify.
log0.1 10x2 = log0.1...
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