An environmental scientist monitors the concentration, C C\,C parts per million (ppm), of a specific chemical pollutant in a reservoir over time t t\,t months. The scientist identifies a linear relationship between log10C\log_{10} Clog10C and ttt. The line representing this relationship passes through the points (2.4,1.82)(2.4, 1.82)(2.4,1.82) and (6.4,0.42)(6.4, 0.42)(6.4,0.42).
Determine the equation for the relationship between log10C\log_{10} Clog10C and ttt.
Hence, or otherwise, express C C\,C in the form kbtkb^tkbt, giving the values of the constants k k\,k and b b\,b to 3 significant figures.
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.