Logarithmic and exponential functions — common mistakes

The formula $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$ is extremely useful and frequently tested. Always identify the 'inside function' $f(x)$, find its derivative $f'(x)$, and place it over the original inside function.

'ln' specifically means 'natural logarithm', which is logarithm to the base $e$ (i.e., $\log_e$). The 'log' button on a calculator usually means logarithm to the base 10 (i.e., $\log_{10}$). In A-Level Pure Mathematics, if you see 'log' written without a base, it often means 'ln', but the notation $\ln$ is preferred to avoid ambiguity.

The function $y = \ln x$ is the inverse of $y = e^x$. The range of $y=e^x$ is all positive real numbers ($y>0$). For an inverse function, the domain and range are swapped. Therefore, the domain of $y = \ln x$ must be the range of $y=e^x$, which is $x>0$. You cannot take the logarithm of a zero or negative number.

To solve for an unknown in an exponent, you should take logarithms of both sides. It's usually easiest to use natural logarithms (ln). For $5^x = 10$, we get $\ln(5^x) = \ln(10)$. Using the power rule for logs, this becomes $x \ln 5 = \ln 10$. Finally, divide to find $x$: $x = \frac{\ln 10}{\ln 5}$.

The base $e$ is used for models involving continuous growth or decay, where the rate of change is proportional to the current amount. This is the standard for most P3 modelling questions involving populations, radioactivity, etc. The general form $N=N_0 e^{kt}$ is provided in many questions. If the growth happens in discrete intervals (e.g., once per year), a model like $P = P_0 (1.05)^t$ might be used, but this is less common in P3.