Positive Logarithms

★ 6/10functions Logarithms inequality Casework

Problem

Define the positive logarithm by

\ln^+ x = \begin{cases} 0, & 0 < x < 1, \\ \ln x, & x \geqslant 1. \end{cases}

Consider the following four propositions, where $a > 0$ and $b > 0$ throughout:

1. $\ln^+\left(a^b\right) = b\ln^+ a$ ; 2. $\ln^+(ab) = \ln^+ a + \ln^+ b$ ; 3. $\ln^+\left(\dfrac{a}{b}\right) \geqslant \ln^+ a - \ln^+ b$ ; 4. $\ln^+(a + b) \leqslant \ln^+ a + \ln^+ b + \ln 2$ .

Determine which of these propositions are true, and justify your answers.

Answer

Solution

Difficulty	6/10
Topics	functions, Logarithms, inequality, Casework

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