Higher Maths: Exponential and Logarithmic Modelling from Data

A question on exponential or logarithmic modelling appears regularly in Higher Mathematics, often in the calculator paper. This material practises the skills listed for it in the Higher Mathematics course specification: solving equations of the forms $\log y=b\log x+\log a$ with $y=a{x}^b$ and $\log y=x\log b+\log a$ with $y=a{b}^x$, using a straight-line graph to confirm such a relationship, solving logarithmic and exponential equations, and modelling situations with the logarithmic or exponential function.

The first group of questions is about the two models. You decide which graph, $\log y$ against $\log x$ or $\log y$ against $x$, should be a straight line for each model, identify what the gradient and the intercept represent, and recover $a$ and $b$ from a line described by its gradient and intercept or by two points on it. The questions use both base-10 and natural logarithms, and the explanations point out the common slip of undoing a natural logarithm with 10 instead of $e$, or of reading the intercept as $a$ itself. Two questions give two pairs of values and ask for $a$ and $b$ directly, by dividing one equation by the other.

The second group solves equations with the laws of logarithms: combining logarithms into one, converting to exponential form and rejecting a root that lies outside the domain of a logarithm, and solving equations such as $e^{2x}=7$ and $3^{2x - 1} = 5^x$ by taking logarithms of both sides.

The last group models growth and decay with $e$: finding the constant in a decay model from a half-life, finding a growth constant from one measurement and using it to predict when a population reaches a given size, and, in the written work, a car losing value in pounds over time.

The category's functions and logarithms material covers the basic laws and simple exponential equations; this one goes on to modelling from data. It offers three formats. The quiz has 12 questions with worked explanations. The flashcards collect the two linear forms, how to read $a$ and $b$ from a graph and the laws of logarithms. The printable written work has 8 longer tasks set out by hand, including fitting both models to data, a proof of the linear form, a half-life and a depreciation problem.

It is aimed at S5 and S6 students preparing for Higher Mathematics. Every value is given in the question, so no graph paper or diagram is needed.

• Convert y = axᵇ and y = abˣ to straight-line form using logarithms
• Decide which model fits data from the graph that is a straight line
• Find a and b from the gradient and intercept of a log graph
• Find a and b from two pairs of corresponding values
• Solve logarithmic equations, rejecting roots outside the domain
• Solve exponential equations by taking logarithms
• Model growth and decay, including half-life, with the exponential function

Practice material written by Zestly, based on the Qualifications Scotland (formerly SQA) Higher Mathematics course specification, version 3.0 (algebraic skills: solving logarithmic and exponential equations; equations of the forms log y = b log x + log a, y = axᵇ and log y = x log b + log a, y = abˣ; straight-line graphs; mathematical modelling with the logarithmic or exponential function).