The mass, m grams, of a radioactive substance is given by the formula m = m 0 e - k t , where t is the time in days after the mass was first recorded and m 0 and k are constants. The table below shows experimental values of t and m . t 10 20 30 40 50 m 40 . 9 33 . 5 27 . 4 22 . 5 18 . 4 The half-life of a radioactive substance is the time it takes to decay to half of its original mass. Find the half-life of this radioactive substance.

Given that the mass, m grams, of a radioactive substance is given by the formula m = m 0 e - k t , where t is the time in days after the mass was first recorded and m 0 and k are constants. The table below shows experimental values of t and m . t 10 20 30 40 50 m 40 . 9 33 . 5 27 . 4 22 . 5 18 . 4 Given exponential function m = m 0 e - k t Taking natural logarithms on both sides gives: ⇒ ln m = ln m 0 e - k t Logarithms with a base e , called natural logarithms and if y = e x then x = ln y Using the product rule of logarithms, ln u v = ln u + ln v ⇒ ln m = ln m 0 + ln e - k t Using the power law of logarithms, ln u m = m ln u ⇒ ln m = ln m 0 - k t ln e ⇒ ln m = ln m 0 - k t × 1 ∵ ln e e = 1 ⇒ ln m = - k t + ln m 0 Comparing the above equation with Y = m X + c , ⇒ Y = ln m , m = - k , X = t and c = ln m 0 Hence, the graph of ln m against t , needs to be drawn using points in the table. t 10 20 30 40 50 ln m 3 . 71 3 . 51 3 . 31 3 . 11 2 . 91 The points form an approximate straight line. ⇒ - k = Gradient of straight line. Using last two points to find the slope of straight line, ⇒ - k = 2 . 91 - 3 . 11 50 - 40 ⇒ - k = - 0 . 2 10 ⇒ k = 0 . 02 To find the intercept, we can use the linear equation ln m = - k t + ln m 0 ⇒ 3 . 71 = - 0 . 02 × 10 + ln m 0 ⇒ ln m 0 = 3 . 91 ⇒ m 0 = e 3 . 91 ⇒ m 0 ≈ 50 (rounded to the nearest integer) Substituting the values in m = m 0 e - k t ⇒ m = 50 e - 0 . 02 t The half-life of a radioactive substance is the time it takes to decay to half of its original mass. We need to find time for original mass is halved i.e., m = m 0 2 ⇒ m 0 2 = m 0 e - 0 . 02 t ⇒ e - 0 . 02 t = 1 2 Taking natural logs of both sides: ⇒ - 0 . 02 t = ln 1 2 ⇒ t = - 1 0 . 02 ln 1 2 ⇒ t = 34 . 7 ⇒ t ≈ 35 days

MEDIUM

AS and A Level

IMPORTANT

Earn 100

Variables $x$ and $y$ are related so that, when $\log_{10} y$ is plotted on the vertical axis and $x$ is plotted on the horizontal axis, a straight-line graph passing through the points $(2, 5)$ and $(6, 11)$ is obtained.

Express $y$ in terms of $x$ , giving your answer in the form $y = a \times 10^{b x}$ .

Important Questions on Logarithmic and Exponential Functions

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 5

Variables $x$ and $y$ are related so that, when $\ln y$ is plotted on the vertical axis and $\ln x$ is plotted on the horizontal axis, a straight-line graph passing through the points $(2, 4)$ and $(5, 13)$ is obtained.

Express $\ln y$ in terms of $y = a^{x}$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 5

Express $y$ in terms of $x$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 6

The variables

x

and

y

satisfy the equation

5^{2 y} = 3^{2 x + 1}

. By taking natural logarithms, show that the graph of

\ln y

against

\ln x

is a straight line, and find the exact value of the gradient of this line and state the coordinates of the point at which the line cuts the

y

-axis.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 7

The mass, $m$ grams, of a radioactive substance is given by the formula $m = m_{0} e^{- k t}$ , where $t$ is the time in days after the mass was first recorded and $m_{0}$ and $k$ are constants.

The table below shows experimental values of $t$ and $m$ .

$t$	$10$	$20$	$30$	$40$	$50$
$m$	$40.9$	$33.5$	$27.4$	$22.5$	$18.4$

Draw the graph of $\ln m$ against $t$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 7

The mass, $m$ grams, of a radioactive substance is given by the formula $m = m_{0} e^{- k t}$ , where $t$ is the time in days after the mass was first recorded and $m_{0}$ and $k$ are constants.

The table below shows experimental values of $t$ and $m$ .

$t$	$10$	$20$	$30$	$40$	$50$
$m$	$40.9$	$33.5$	$27.4$	$22.5$	$18.4$

Use your graph to estimate the value of $m_{0}$ and $k$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 7

The mass, $m$ grams, of a radioactive substance is given by the formula $m = m_{0} e^{- k t}$ , where $t$ is the time in days after the mass was first recorded and $m_{0}$ and $k$ are constants.

The table below shows experimental values of $t$ and $m$ .

$t$	$10$	$20$	$30$	$40$	$50$
$m$	$40.9$	$33.5$	$27.4$	$22.5$	$18.4$

The half-life of a radioactive substance is the time it takes to decay to half of its original mass. Find the half-life of this radioactive substance.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 8

The temperature, $T ° C$ , of a hot drink, $t$ minutes after it is made, can be modelled by the equation $T = 25 + k e^{- n t}$ , where $k$ and $n$ are constants. The table below shows experimental values of $t$ and $T$ .

$t$	$2$	$4$	$6$	$8$	$10$
$T$	$63.3$	$57.7$	$52.8$	$48.7$	$45.2$

Convert the equation to a form suitable for drawing a straight-line graph.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>EXERCISE 2H>Q 8

$t$	$2$	$4$	$6$	$8$	$10$
$T$	$63.3$	$57.7$	$52.8$	$48.7$	$45.2$

Draw the straight-line graph and use it to estimate the value of $k$ and $n .$

Variables x and y are related so that, when log10y is plotted on the vertical axis and x is plotted on the horizontal axis, a straight-line graph passing through the points (2, 5) and (6, 11) is obtained. Express y in terms of x, giving your answer in the form y=a×10bx.

Important Questions on Logarithmic and Exponential Functions

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