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Fungrim entry: 77aa12

log ⁣(x+a)log ⁣(x)log ⁣(1+axa)\left|\log\!\left(x + a\right) - \log\!\left(x\right)\right| \le \log\!\left(1 + \frac{\left|a\right|}{x - \left|a\right|}\right)
Assumptions:xRandaRanda0anda<xx \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \ge 0 \,\mathbin{\operatorname{and}}\, \left|a\right| < x
TeX:
\left|\log\!\left(x + a\right) - \log\!\left(x\right)\right| \le \log\!\left(1 + \frac{\left|a\right|}{x - \left|a\right|}\right)

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \ge 0 \,\mathbin{\operatorname{and}}\, \left|a\right| < x
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("77aa12"),
    Formula(LessEqual(Abs(Sub(Log(Add(x, a)), Log(x))), Log(Add(1, Div(Abs(a), Sub(x, Abs(a))))))),
    Variables(x, a),
    Assumptions(And(Element(x, RR), Element(a, RR), GreaterEqual(a, 0), Less(Abs(a), x))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-15 11:00:55.020619 UTC