Exponents and logarithms: exam review Exam review: second paper One question at a time

Use the calculator where it helps. [7]

1. (a) Solve \(10^x = 3600\). Give x to 3 significant figures. [2]

   • M1 for x = log10 3600
   • A1 3.56

2. (b) Solve \(\log_{10}(4x + 1) = 2\). Give the exact value of x. [2]

   • M1 for 4x + 1 = 10^2 = 100
   • A1 x = \frac{99}{4} (24.8)

3. (c) Solve \(2 + 4\ln x = 11\). Give the exact value of x, and x to 3 significant figures. [3]

   • M1 for isolating the log: ln x = \frac{9}{4}
   • M1 for x = e^(\frac{9}{4})
   • A1 9.49

Do not use a calculator for this question. [5]

1. (a) Write \(\dfrac{(2x^5)^3}{12x^{-4}}\) in the form \(ax^p\). Write down the values of a and p. [3]

   • M1 for (2x^5)^3 = 8x^15
   • M1 for dividing by x^(−4), so the indices are added: 15 + 4
   • A1 a = 2/3, p = 19

2. (b) Find the exact value of \(\left(\dfrac{2}{5}\right)^{-3}\). [2]

   • M1 for turning the fraction over: (5/2)^3
   • A1 125/8

Do not use a calculator for this question. [5]

1. (a) Find the exact value of \(\left(\frac{3}{5}\right)^{-3}\). [2]

   • M1 for flipping the fraction: \(\left(\frac{5}{3}\right)^{3}\)
   • A1 \(\frac{125}{27}\)

2. (b) Write \(\dfrac{(3x^4)^2}{18x^{-3}}\) in the form \(ax^p\). Write down the values of a and p. [3]

   • M1 for \((3x^4)^2=9x^{8}\)
   • M1 for dividing the numbers and subtracting the powers
   • A1 \(a = \frac{1}{2}\), p = 11

Solve each equation. [5]

1. (a) Solve \(\ln(x-2)=1.5\). Give x to 3 significant figures. [2]

   • M1 for \(x-2=e^{1.5}\)
   • A1 \(x = e^{1.5}+2 = 6.48\)

2. (b) Solve \(4^{x+1}=32^{x-2}\). Give the exact value of x. [3]

   • M1 for writing both sides as powers of 2: \(2^{2(x+1)}=2^{5(x-2)}\)
   • M1 for \(2x+2=5x-10\)
   • A1 x = 4

A water tank holds \(7.2\times10^{9}\) litres. A swimming pool holds \(3.0\times10^{5}\) litres. [5]

1. (a) Find how many pools the water in the tank would fill. Give your answer in the form \(a\times10^{k}\), where \(1\le a<10\) and \(k\in\mathbb{Z}\). [2]

   • M1 for \(\dfrac{7.2\times10^{9}}{3.0\times10^{5}}\) (\(2.4\times10^{4}\), the a values divided and the indices subtracted)
   • A1 \(2.4\times10^{4}\)

2. (b) A second tank holds \(9.6\times10^{8}\) litres. Find the total number of litres the two tanks hold. Give your answer in the form \(a\times10^{k}\). [2]

   • M1 for writing both numbers with the same power of 10, for example \(7.2\times10^{9}+0.96\times10^{9}\)
   • A1 \(8.16\times10^{9}\)

3. (c) One litre is \(10^{3}\) millilitres. Write down the volume of the first tank in millilitres, in the form \(a\times10^{k}\). [1]

   • A1 \(7.2\times10^{12}\) (\(7.2\times10^{9}\times10^{3}\))

The intensity I, in watts per square metre, of a sound at a level of L decibels satisfies \(\log_{10}I=-12+0.1L\). [5]

1. (a) Find the intensity of a sound at a level of 85 decibels. Give your answer in the form \(a\times10^{k}\), where \(1\le a<10\) and \(k\in\mathbb{Z}\), with a correct to 3 significant figures. [2]

   • M1 for \(\log_{10}I=-12+0.1\times85=-3.5\), so \(I=10^{-3.5}\)
   • A1 \(3.16\times10^{-4}\)

2. (b) A sound at 110 decibels and a sound at 80 decibels are compared. Find how many times more intense the 110 decibel sound is. Give your answer to 3 significant figures. [3]

   • M1 for \(\log_{10}I_{110}-\log_{10}I_{80}=0.1\times(110-80)=3\) (the \(-12\) cancels)
   • M1 for \(\log_{10}\dfrac{I_{110}}{I_{80}}=3\), so \(\dfrac{I_{110}}{I_{80}}=10^{3}\)
   • A1 1000

The area N (m²) of a pond covered by algae, t days after it is first measured, is modelled by \(N = 30\times1.15^{t}\). [8]

1. (a) Write down the value of N when t = 0, and find the value of N when t = 5. [2]

   • A1 N = 30
   • A1 60.3 (30 × 1.15^5)

2. (b) Use your calculator to find when N reaches 150. Give t to 3 significant figures. [2]

   • M1 for solving 30 × 1.15^t = 150
   • A1 11.5

3. (c) A second model is \(M = 90\times1.05^{t}\). Find the time when M = N. Give t to 3 significant figures. [2]

   • M1 for solving 30 × 1.15^t = 90 × 1.05^t
   • A1 12.1

4. (d) Find the time for N to multiply by 10. Give t to 3 significant figures. [2]

   • M1 for solving 1.15^t = 10
   • A1 16.5

The concentration C mg/L of a drug in a patient's blood, t hours after a dose, is modelled by \(C=60\times0.78^{t}\). [7]

1. (a) Write down the value of C when \(t=0\), and find the value of C when \(t=5\). Give your answer to 3 significant figures. [2]

   • A1 60
   • A1 17.3 (\(60\times0.78^{5}\))

2. (b) Write down the percentage of the C lost each hour. [1]

   • A1 22% (0.78 means a decrease of 22%)

3. (c) Use your calculator to find when \(C=25\). Give your answer to 3 significant figures. [2]

   • M1 for solving \(60\times0.78^{t}=25\)
   • A1 t = 3.52

4. (d) The drug is safe once the concentration is less than 2 mg/L. Find the first whole number of hours after which it is safe. [2]

   • M1 for solving \(60\times0.78^{t}=2\) (t = 13.7)
   • A1 14 (the first whole number above 13.7)