Exponentials and logarithms — WMA13 IAL Mathematics practice questions · Study-IAL

Exponentials and logarithms

WMA13Topic 3.33 questions

The functions $e^x$ and $\ln x$ , and using logarithmic graphs to estimate parameters.

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Original practice questions written in the style of the exam.

Question 16 marks
Two variables are believed to satisfy $y = ax^n$ , where $a$ and $n$ are constants.

A graph of $\log_{10} y$ against $\log_{10} x$ is a straight line passing through the points $(0.5, 1.449)$ and $(2, 3.699)$ .

(a) Find the values of $n$ and $a$ , giving $a$ to the nearest integer.

(b) Hence estimate $y$ when $x = 4$ .
Mark scheme
- B1 $\log_{10} y = \log_{10} a + n\log_{10} x$ (1)
Question 25 marks
The temperature, $T\,^\circ\text{C}$ , of a cup of tea $t$ minutes after it is made is modelled by

$T = 20 + 75e^{-kt}$

nlog10x

M1 Gradient

= \dfrac{3.699 - 1.449}{2 - 0.5}

(1)

n = 1.5

(1)

M1 Finds intercept

\log_{10} a = 0.699

and uses

a = 10^{0.699}

(1)

a = 5

(1)

y = 5 \times 4^{1.5} = 40

(1)

75e−kt

where $k$ is a positive constant.

(a) State the temperature of the tea when it is made.

Ten minutes after it is made, the temperature of the tea is $50\,^\circ\text{C}$ .

(b) Find the exact value of $k$ .

Mark scheme

7.5