y = ln(x) and its Tangent

$\dfrac{d}{dx}\ln x = \dfrac{1}{x}$ for $x > 0$. So at $x_0 = 2$, the slope is $0.5$, and at $x_0 = 1$ the slope is $1$. This is one of the most-used derivatives in calculus.

Slide $x_0 > 0$ and the lab plots $y = \ln x$ with the tangent line $y = \ln(x_0) + (x - x_0)/x_0$. It reports $f(x_0) = \ln(x_0)$ and $f'(x_0) = 1/x_0$. At $x_0 = 1$, $\ln 1 = 0$ and slope $1$, so the tangent is $y = x - 1$.

Because the natural log uses base $e$, defined so that $\dfrac{d}{dx}e^x = e^x$. As inverses, $\ln$ and $e^x$ share clean derivative formulas. For other bases, $\dfrac{d}{dx}\log_a x = \dfrac{1}{x\ln a}$ — the extra $\ln a$ is a JEE memory hook.

Information theory's Shannon entropy $-\sum p_i\ln p_i$ has derivative pieces $-1 - \ln p_i$. Sound intensity in decibels uses $\log_{10}$, and derivatives there follow the same pattern. Chemical pH and earthquake Richter scale calculations also involve log derivatives indirectly.