Edexcel A-Level Maths: Exponentials and Logarithms

A mug of coffee is left to cool in a room. The temperature $T$T of the coffee, measured in degrees Celsius, is modelled by

$T = 20 + A\,e^{-kt}$T=20+Ae−kt

where $t$t is the time in minutes after the coffee is poured, and $A$A and $k$k are positive constants. At the instant the coffee is poured its temperature is $90\,^{\circ}\text{C}$90∘C, and $10$10 minutes later its temperature has fallen to $55\,^{\circ}\text{C}$55∘C.

(a) Use the information given to find the value of $A$A and show that $k = \dfrac{1}{10}\ln 2$k=101​ln2. (4 marks)

(b) Using your model, find: (i) the temperature of the coffee $25$25 minutes after it is poured, giving your answer to the nearest degree; (ii) the time, to the nearest minute, at which the temperature of the coffee first falls to $30\,^{\circ}\text{C}$30∘C. (5 marks)

(c) Use the model to explain what happens to the temperature of the coffee in the long term, and state, with a reason, whether the coffee cools more quickly at $t = 0$t=0 or at $t = 10$t=10. (3 marks)

A biologist measures the surface area $y\ \text{cm}^2$y cm2 of a growing leaf when its length is $x\ \text{cm}$x cm. She believes the data can be modelled by

$y = a x^{n},$y=axn,

where $a$a and $n$n are constants. Her readings are shown in the table.

Length $x$x (cm)	4	9	25	36
Area $y$y (cm$^2$2)	24.0	81.0	375	648

(a) Show that if the model $y = ax^{n}$y=axn holds, then a graph of $\log_{10} y$log10​y against $\log_{10} x$log10​x should give a straight line, and state how the gradient and the vertical intercept of this line are related to $a$a and $n$n. (3 marks)

(b) The biologist plots $\log_{10} y$log10​y against $\log_{10} x$log10​x and finds that the points lie close to a straight line passing through $(0.50,\ 1.23)$(0.50, 1.23) and $(1.50,\ 2.73)$(1.50, 2.73). Use this line to find the value of $n$n and the value of $a$a, each to $2$2 significant figures. (5 marks)

(c) Use your values of $a$a and $n$n to estimate the surface area of a leaf of length $16\ \text{cm}$16 cm. (2 marks)

(a) Solve the equation

$e^{2x} - 7e^{x} + 12 = 0,$e2x−7ex+12=0,

giving your answers as exact values in terms of natural logarithms. (4 marks)

(b) Solve the equation

$\ln x + \ln(x - 3) = \ln 4,$lnx+ln(x−3)=ln4,

showing clearly why any solution you discard is not valid. (4 marks)

Solve the equation

$\log_{2}(x + 3) + \log_{2}(x - 1) = 5,$log2​(x+3)+log2​(x−1)=5,

giving any value(s) of $x$x you reject, with a reason. (6 marks)

Solve the equation

$5^{\,2x - 1} = 30.$52x−1=30.

Give your answer as an exact value in terms of natural logarithms, and also as a decimal correct to $3$3 significant figures. (5 marks)

Without using a calculator, show that

$2\log_{5} 10 - \log_{5} 4 = 2.$2log5​10−log5​4=2.

You must show your use of the laws of logarithms. (4 marks)