The exponential function and its derivative, the natural logarithm, the laws of logarithms, solving exponential and logarithmic equations, and using logarithms to linearise data and model exponential growth and decay.

What this dot point is asking

AQA wants you to use the exponential function with base $e$ and the natural logarithm, apply the laws of logarithms, solve exponential and logarithmic equations, use logarithms to convert relationships into a straight line, and set up and interpret exponential growth and decay models. Modelling questions, where you read off or interpret the constants in $N = N_0 e^{kt}$, are a reliable source of marks.

The number e and the natural logarithm

The constant $e \approx 2.718$ is defined so that the curve $y = e^x$ has gradient equal to its own height at every point, which makes $\frac{d}{dx}(e^x) = e^x$ and, with the chain rule, $\frac{d}{dx}(e^{kx}) = ke^{kx}$. The inverse function is the natural logarithm $\ln x = \log_e x$, so the two undo each other: $e^{\ln x} = x$ for $x > 0$, and $\ln(e^x) = x$ for all $x$. The graph of $y = e^x$ passes through $(0, 1)$ and rises steeply; $y = \ln x$ passes through $(1, 0)$ and is its reflection in $y = x$.

Laws of logarithms

The power law is the workhorse for equations: it pulls an unknown exponent down to the front, where it can be isolated. The product and quotient laws let you combine several logarithms into one before exponentiating.

Solving equations

Equations like $3^{2x} - 10(3^x) + 9 = 0$ are hidden quadratics: substitute $y = 3^x$ to get $y^2 - 10y + 9 = 0$, solve, then recover $x$. For logarithmic equations, combine the logs, exponentiate, solve, and reject any solution that makes a log argument non-positive.

Linearising data and modelling

If $y = ax^n$, then $\log y = \log a + n\log x$, so plotting $\log y$ against $\log x$ gives a straight line of gradient $n$ and intercept $\log a$. If $y = ab^x$, then $\log y = \log a + x\log b$, so plotting $\log y$ against $x$ gives gradient $\log b$ and intercept $\log a$. Identifying which axes to log, then reading the gradient and intercept, is the standard exam route to finding the model constants. The distinction matters: a power law $y = ax^n$ becomes linear on a log-log plot ($\log y$ against $\log x$), whereas an exponential law $y = ab^x$ becomes linear on a log-linear plot ($\log y$ against $x$). Spotting which one fits given data is itself an examined skill, and the gradient then gives either the power $n$ or $\log b$, while the intercept gives $\log a$.

To recover the constants, exponentiate the intercept. For a log-linear fit $\log y = (\log b)x + \log a$, the intercept $c$ gives $a = 10^c$ (or $e^c$ for natural logs) and the gradient $m$ gives $b = 10^m$. Always state which base of logarithm you used, because mixing base $10$ and base $e$ when back-substituting is a common and costly slip.