The market value VVV, in pounds (£), of a limited-edition bronze sculpture is modelled by the equation
V=pqt V = pq^t V=pqtwhere ttt is the number of years since its initial release and ppp and qqq are constants.
A researcher records the market value over several years. The graph of log10V\log_{10} Vlog10V against ttt is a straight line that passes through the points (0,3.15)(0, 3.15)(0,3.15) and (12,5.07)(12, 5.07)(12,5.07).
Determine an equation for this line in the form log10V=mt+c\log_{10} V = mt + clog10V=mt+c.
Calculate the value of ppp and the value of qqq, giving your answers to 3 significant figures.
Use the model to estimate the number of years, ttt, until the sculpture reaches a value of £50,000.
104 exam-style questions on OCR A Level Maths 1.6 Exponentials and Logarithms, covering 1.6.1 Properties of the exponential function, 1.6.2 Gradient of e^(kx) (A-level only), 1.6.3 Definition of the logarithm, 1.6.4 The natural logarithm function, 1.6.5 ln x as inverse of e^x, 1.6.6 Laws of logarithms, 1.6.7 Equations involving exponentials, 1.6.8 Reduction to linear form, and 1.6.9 Modelling using exponential functions. Each one has a worked solution and a mark scheme showing where the marks go.