Fungrim entry: 77aa12

\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:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \ge 0 \,\mathbin{\operatorname{and}}\, \left|a\right| \lt 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| \lt x

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Log`	$\log\!\left(z\right)$	Natural logarithm
`RR`	$\mathbb{R}$	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))))

Topics using this entry

Natural logarithm