OCR AS Maths Exponentials and logarithms J"integral Topic assessment 1. Write logx-3log2y + 4log = as a single logarithm. [2] 2. Express log in terms of log p, log q and log r. [2] 3. Solve the following equations; (a) 32x =5 [2] (b) 2e" =3e"* + 5 [3] (c) In(2r+1) = In.x+2 [3] giving your answers in exact form. The temperature T'C of the water in a kettle t minutes after boiling is modelled by the equation T - 20 + 80e-0.5t (a) What is the initial temperature of the water? (b) Find the temperature of the water after 5 minutes. [2] (c) Find the time at which the temperature of the water is 30.C. [3] (d) Find the initial rate of cooling, and the rate of cooling after 2 minutes. [3] (e) What will the long-term temperature of the water be? [1] 5. In an experiment, the number of bacteria, N, in a culture was estimated at time t days after he experiment started. The results were as follows: 2 3 4 5 6 N 120 170 250 400 620 910 It is believed that the relationship between N and t can be expressed in the form N = ab' where a and b are constants. (a) Use logarithms to draw a linear graph. [3] (b) Use your graph to estimate the values of a and b. [3] (c) Estimate the number of bacteria present after 20 days. State, with a reason, whether your estimate is likely to be a good one. [2] 6. The equation log, (2x + p) =2log, (2x) -1 has one real root. What is the value of p? [5] MEI Mathematics Education 1 of 2 28/10/20 @ MEI Innovation integralmaths.org