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Unformatted text preview: to get 6. What are we doing with this exponent?
We are putting it on 117. By deﬁnition we get 6. In other words, the exponential function
f (x) = 117x undoes the logarithmic function g (x) = log117 (x).
Up until this point, restrictions on the domains of functions came from avoiding division by zero
and keeping negative numbers from beneath even radicals. With the introduction of logs, we now
have another restriction. Since the domain of f (x) = logb (x) is (0, ∞), the argument8 of the log
must be strictly positive.
Example 6.1.4. Find the domain of the following functions. Check your answers graphically using
1. f (x) = 2 log(3 − x) − 1
2. g (x) = ln x
1. We set 3−x > 0 to obtain x < 3, or (−∞, 3). The graph from the calculator below veriﬁes this.
Note that we could have graphed f using transformations. Taking a cue from Theorem 1.7, we
rewrite f (x) = 2 log10 (−x + 3) − 1 and ﬁnd the main function involved is y = h(x) = log10 (x).
We select three points to track, 10 , −1 ,...
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