The variables xxx and yyy are connected by the relationship
y=10−15x2.5,x>0 y = \frac{10^{-15}}{x^{2.5}}, \quad x > 0 y=x2.510−15,x>0Sketch the graph of log10y\log_{10} ylog10y against log10x\log_{10} xlog10x. Clearly label the coordinates of the points of intersection of the graph with the coordinate axes.
A population of microbes NNN increases over time ttt hours according to the model N=abtN = ab^tN=abt, where aaa and bbb are positive constants. A scientist plots the linear relationship between log9N\log_{9} Nlog9N and ttt. The resulting line has a positive gradient and passes through the points (0,2.5)(0, 2.5)(0,2.5) and (−10,0)(-10, 0)(−10,0). Determine the exact values of aaa and bbb.
104 exam-style questions on OCR (MEI) A Level Maths 1.8 Exponentials and Logarithms, covering 1.8.1 The function y = a^x, 1.8.2 Convert between index and logarithmic form, 1.8.3 Logarithm as inverse of exponential, 1.8.4 Laws of logarithms, 1.8.5 Values of log_a a and log_a 1, 1.8.6 Solve equations of the form a^x = b, 1.8.7 Reduce y = ax^n and y = ab^x to linear form, 1.8.8 The function y = e^x, 1.8.9 Gradient of e^kx (A-level only), 1.8.10 The function y = ln x, and 1.8.11 Exponential growth and decay. Each one has a worked solution and a mark scheme showing where the marks go.