https://tvmaaren.nl/mathematics/Recent content in Mathematics on Thomas van MaarenHugoen-usThu, 22 Jan 2026 20:44:19 +0100https://tvmaaren.nl/mathematics/logarithms/Thu, 22 Jan 2026 20:44:19 +0100https://tvmaaren.nl/mathematics/logarithms/<p>Hello, I would like to define things related to logarithms: \(\ln,e\). I want to then prove facts like \(\ln(e)=1\), that \(\ln\) is the inverse of \(x\mapsto e^x\) and that the derivative of \(\ln\) is \(x\mapsto 1/x\).</p> <p><strong>Definition</strong> For \(0 < x\) we define </p> \[\ln(x) := \lim_{n\to \infty} n(\sqrt[n]{x}-1).\]<p>I should prove that this that the sequence convergence, that it is continuous and that it is diffferentiable, but I don&rsquo;t feel like doing this at the moment.</p>