Exponents and logarithms: exam review Exam review: second paper

Use the calculator where it helps. [7]

1. (a) Solve \(10^x = 3600\). Give x to 3 significant figures. [2]
2. (b) Solve \(\log_{10}(4x + 1) = 2\). Give the exact value of x. [2]
3. (c) Solve \(2 + 4\ln x = 11\). Give the exact value of x, and x to 3 significant figures. [3]

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

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

1. (a) Find the exact value of \(\left(\frac{3}{5}\right)^{-3}\). [2]
2. (b) Write \(\dfrac{(3x^4)^2}{18x^{-3}}\) in the form \(ax^p\). Write down the values of a and p. [3]

Solve each equation. [5]

1. (a) Solve \(\ln(x-2)=1.5\). Give x to 3 significant figures. [2]
2. (b) Solve \(4^{x+1}=32^{x-2}\). Give the exact value of x. [3]

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]
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]
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]

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]
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]

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]
2. (b) Use your calculator to find when N reaches 150. Give t to 3 significant figures. [2]
3. (c) A second model is \(M = 90\times1.05^{t}\). Find the time when M = N. Give t to 3 significant figures. [2]
4. (d) Find the time for N to multiply by 10. Give t to 3 significant figures. [2]

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]
2. (b) Write down the percentage of the C lost each hour. [1]
3. (c) Use your calculator to find when \(C=25\). Give your answer to 3 significant figures. [2]
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]