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**Unformatted text preview: **to inequalities involving logarithmic functions. Since logarithmic functions
are continuous on their domains, we can use sign diagrams.
Example 6.4.2. Solve the following inequalities. Check your answer graphically using a calculator.
2 Recall that an extraneous solution is an answer obtained analytically which does not satisfy the original equation. 6.4 Logarithmic Equations and Inequalities
1. 1
≤1
ln(x) + 1 371 2. (log2 (x))2 < 2 log2 (x) + 3 3. x log(x + 1) ≥ x Solution.
1. We start solving 1
ln(x)+1 1
ln(x)+1 − 1
−
reduces to ln(ln(x)
x)+1 ≤ 1 by getting 0 on one side of the inequality: 1
Getting a common denominator yields ln(x)+1
ln(x)
ln(x)
or ln(x)+1 ≥ 0. We deﬁne r(x) = ln(x)+1 and − ln(x)+1
ln(x)+1 ≤ 0 which ≤ 0.
≤ 0, set about ﬁnding the domain and the zeros
of r. Due to the appearance of the term ln(x), we require x > 0. In order to keep the
denominator away from zero, we solve ln(x) + 1 = 0 so ln(x) = −1, so x = e−1 = 1 . Hence,
e
ln(x)
the domain of r is 0, 1 ∪ 1 , ∞ . To ...

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