Browse Exponentials & logarithms

134 questions at your level

DifficultyAllLevel 1Level 2Level 3Level 4Level 5

The laws of logarithms

45 questions

LessonNot started
Using logarithms, solve the equation

72x−1=3x+4,7^{2x - 1} = 3^{x + 4},

giving your answer correct to 3 significant figures.

(3 marks)
●●●●●Level 23 marksStart
Figure 1(−4.5, 0)(0, 3)Olog₁₀ xlog₁₀ yFigure 1 shows a linear relationship between log⁡10y\log_{10} y and log⁡10x\log_{10} x

The line passes through the points (−4.5,0)(-4.5, 0) and (0,3)(0, 3) as shown.
(a) Find an equation linking log⁡10y\log_{10} y with log⁡10x\log_{10} x

(2 marks)
(b) Hence, or otherwise, express yy in the form pxqpx^q where pp and qq are rational constants.

(3 marks)
●●●●●Level 25 marksStart
Solve the equation

log⁡2(x+5)+log⁡2(x−1)=4\log_2(x+5) + \log_2(x-1) = 4. (4 marks)
●●●●●Level 24 marksStart

Solving exponential and logarithmic equations

57 questions

LessonNot started
Using logarithms, solve the equation

72x−1=3x+4,7^{2x - 1} = 3^{x + 4},

giving your answer correct to 3 significant figures.

(3 marks)
●●●●●Level 23 marksStart
Solve the equation

log⁡2(x+5)+log⁡2(x−1)=4\log_2(x+5) + \log_2(x-1) = 4. (4 marks)
●●●●●Level 24 marksStart
Solve the equation

8x+1=42x−18^{x+1} = 4^{2x-1}

(4 marks)
●●●●●Level 24 marksStart
Every hour, an inspector takes a random sample of 33 bolts from a production line and records the range of their lengths. The probability that the range of a sample is 00 is 0.180.18, independently from hour to hour.

The inspector takes nn samples. The random variable YY represents the number of these nn samples that have a range of 00.

Calculate the minimum value of nn such that P(Y=0)<0.02P(Y = 0) < 0.02

(3 marks)
●●●●●Level 23 marksStart

Exponential growth and decay models

48 questions

LessonNot started
A metal casting is removed from a furnace and left to cool in a workshop.

The temperature of the casting, H ∘H\ ^\circC, tt minutes after it is removed from the furnace, is modelled by the equation

H=A e−Bt+18H = A\,e^{-Bt} + 18

where AA and BB are constants.

Initially, the temperature of the casting was 618 ∘618\ ^\circC.
(a) State the value of AA.

(1 mark)

Initially, the casting was cooling at a rate of 32 ∘32\ ^\circC per minute.
(b) Find a complete equation linking HH and tt, giving the value of BB to 3 decimal places.

(3 marks)
●●●●●Level 24 marksStart
A blacksmith takes a horseshoe out of a forge and places it on the stone floor of a workshop. The temperature of the workshop is a constant 15 ∘C15\,^\circ\text{C}

At time tt minutes after the horseshoe is taken out of the forge, the temperature of the horseshoe, θ ∘C\theta\,^\circ\text{C}, decreases at a rate which is proportional to θ−15\theta - 15

When the horseshoe is taken out of the forge, its temperature is 815 ∘C815\,^\circ\text{C} and its temperature is decreasing at a rate of 60 ∘C60\,^\circ\text{C} per minute.

Show that

dθdt=−0.075(θ−15)\frac{d\theta}{dt} = -0.075(\theta - 15)

(3 marks)
●●●●●Level 23 marksStart

Logarithms and non-linear data

20 questions

LessonNot started
Figure 1(−4.5, 0)(0, 3)Olog₁₀ xlog₁₀ yFigure 1 shows a linear relationship between log⁡10y\log_{10} y and log⁡10x\log_{10} x

The line passes through the points (−4.5,0)(-4.5, 0) and (0,3)(0, 3) as shown.
(a) Find an equation linking log⁡10y\log_{10} y with log⁡10x\log_{10} x

(2 marks)
(b) Hence, or otherwise, express yy in the form pxqpx^q where pp and qq are rational constants.

(3 marks)
●●●●●Level 25 marksStart