Quick questions on Exponential and logarithmic functions (TCE Mathematics Methods, Tasmania)

Calculators evaluate $\ln$ and $\log_{10}$, but exam questions can use any base. The change-of-base rule $\log_{a}x = \dfrac{\ln x}{\ln a}$ converts to a base you can evaluate. For example, $\log_{5}40 = \dfrac{\ln 40}{\ln 5} = \dfrac{3.6889}{1.6094} \approx 2.292$. The same rule explains why $\dfrac{d}{dx}\log_{a}x = \dfrac{1}{x\ln a}$: writing $\log_{a}x = \dfrac{\ln x}{\ln a}$ and differentiating the constant multiple $\dfrac{1}{\ln a}$ of $\ln x$ gives the result directly.