Edexcel A-Level Mathematics AS P2.5 Exponentials and Logarithms Questions

Practise exponential graphs, logarithm laws and log-based equation solving, from sketch features to exact and decimal answers.

Exam points

sketch exponential curves, showing intercepts and horizontal asymptotes
use log laws to turn logarithmic equations into algebraic equations
take logarithms to solve exponential models and interpret constants in context

Covered syllabus points

P2.5.1 - y = ax and its graph P2.5.2 - Laws of logarithms P2.5.3 - Solving equations using logarithms

[Maximum number: 3]

Sketch the curve with equation

y=a^{-x}+4

where a is a constant and a>1
On your sketch show
- the coordinates of the point of intersection of the curve with the y-axis
- the equation of the asymptote to the curve.

[Maximum number: 8]

(a)

Given that

3 \log _{3}(2 x-1)=2+\log _{3}(14 x-25)

show that

2 x^{3}-3 x^{2}-30 x+56=0

[ 4 ]

Official answer figure

$2=\log_3 3^2$ $3\log_3(2x-1)=\log_3(2x-1)^3$

or

$\log_3 3^2+\log_3(14x-25)=\log_3(3^2(14x-25))$ $(2x-1)^3=9(14x-25)$ $8x^3-12x^2+6x-1=126x-225$ $2x^3-3x^2-30x+56=0$

A1*

(b)

Hence, using algebra and showing each step of your working, solve

$3\log_3(2x-1)=2+\log_3(14x-25)$

[ 4 ]

Official answer figure

$2x^3-3x^2-30x+56=0$ $(x+4)(2x^2-11x+14)=0$ (x+4)(2x-7)(x-2)=0
The equation is not defined for x=-4, so the solutions are

$x=2,\quad x=\frac72$

[Maximum number: 3]

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Hence solve

2 \times 4^{x}-2^{x+3}=17 \times 2^{x-1}-4

$4p^2-33p+8=0\Rightarrow(4p-1)(p-8)=0$ $p=\frac14$ or 8 $2^x=\frac14\Rightarrow x=-2;\quad 2^x=8\Rightarrow x=3$ x=-2, 3

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3