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Unformatted text preview: logs ﬁrst with the Power Rule, then deal with addition and subtraction using the Product and
Quotient Rules, respectively. Additionally, we ﬁnd that using log properties in this fashion can
increase the domain of the expression. For example, we leave it to the reader to verify the domain
of f (x) = log3 (x−1)−log3 (x+1) is (1, ∞) but the domain of g (x) = log3 x+1 is (−∞, −1)∪(1, ∞).
We will need to keep this in mind when we solve equations involving logarithms in Section 6.4 - it
is precisely for this reason we will have to check for extraneous solutions.
The two logarithm buttons commonly found on calculators are the ‘LOG’ and ‘LN’ buttons which
correspond to the common and natural logs, respectively. Suppose we wanted an approximation to
log2 (7). The answer should be a little less than 3, (Can you explain why?) but how do we coerce
the calculator into telling us a more accurate answer? We need the following theorem.
Theorem 6.7. (Change of Base) Let a, b > 0, a, b = 1.
• ax = bx logb (a) for all real numbers x.
• loga (x) = logb (x)
for all real numbers x > 0.
logb (a) The proofs of the Change of Base formulas are a result of the other properties studied in...
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