Unformatted text preview: ule for exponential functions given in Theorem 6.5,
f (u + w) = f (u)f (w), says that adding inputs results in multiplying outputs. Hence, whatever f −1
is, it must take the products of outputs from f and return them to the sum of their respective inputs.
Since the outputs from f are the inputs to f −1 and vice-versa, we have that that f −1 must take
products of its inputs to the sum of their respective outputs. This is precisely what the Product Rule
for Logarithmic functions states in Theorem 6.6: g (uw) = g (u) + g (w). The reader is encouraged to
view the remaining properties listed in Theorem 6.6 similarly. The following examples help build
familiarity with these properties. In our ﬁrst example, we are asked to ‘expand’ the logarithms.
This means that we read the properties in Theorem 6.6 from left to right and rewrite products
inside the log as sums outside the log, quotients inside the log as diﬀerences outside the log, and
powers inside the log as factors outside the log. While it is the opposite p...
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