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. [7]

1. (a) Find the exact value of \(64^{-\frac{2}{3}}\). [2]
2. (b) Given that \(y=\dfrac{3}{x^3\sqrt{x}}\), write \(y^2\) in the form \(ax^k\). Write down the values of a and k. [3]
3. (c) Write \(\dfrac{x^3 + 4\sqrt{x}}{x}\) in the form \(ax^p + bx^q\). [2]

Solve each equation. Do not use a calculator. [5]

1. (a) Solve \(x^{\frac{3}{2}} = 125\). [2]
2. (b) Solve \(\left(\sqrt{3}\right)^x = 27^{x-2}\). Give the exact value of x. [3]

Give exact answers. Do not use a calculator. [5]

1. (a) Let \(p=\log_{10}a\) and \(q=\log_{10}b\). Write \(\log_{10}\left(\dfrac{1000\sqrt{a}}{b^{2}}\right)\) in terms of p and q. [3]
2. (b) Write \(3 \ln 2 + \ln 15 − \ln 6\) as a single logarithm in the form \(\ln k\). [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]

Solve each equation. [13]

1. (a) Solve \(\log_{10}(x+2) + \log_{10}4 = 2\). Give an exact answer. [3]
2. (b) Solve \(\ln(3x+2) - \ln(x-3) = 2\). Give x to 3 significant figures. [4]
3. (c) Solve \(3^{x+2} = 8^{x-1}\). Give x to 3 significant figures. [4]
4. (d) Solve \(\log_{x}125 = \frac{3}{2}\). [2]

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]

An ecologist studies how the number of species y found on an island depends on its area x (km²). The areas in the study range from 0.02 km² to 20 000 km². [13]

1. (a) Write down the logarithm to base 10 of each of the values 0.04, 15, 8000. Give each to 3 significant figures. [2]
2. (b) Give one reason why a logarithmic scale is useful for data like this. [1]
3. (c) For the data in the table, the correlation coefficient for \(\log_{10}y\) against x is r = 0.7615, and for \(\log_{10}y\) against \(\log_{10}x\) it is r = 0.9983. Decide whether an exponential model or a power model fits the data better. Give a reason. [2]
4. (d) The line of best fit for \(\log_{10}y\) against \(\log_{10}x\) has gradient 0.298 and intercept 0.708. Write the model in the form \(y=a x^k\). [3]
5. (e) Use the model to estimate y when x = 250. [2]
6. (f) A different relationship has the form \(y=a\times b^{x}\). The graph of \(\log_{10}y\) against x is a straight line through \((0, 0.7)\) and \((3, 1.9)\). Find a and b. [3]

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]

A yeast culture grows so that its mass M, in milligrams, after t hours is modelled by \(M=M_0e^{kt}\), where \(M_0\) and k are constants. At \(t=0\) the mass is 800 mg. At \(t=5\) the mass is 2600 mg. [8]

1. (a) Find the value of k. Give your answer to 3 significant figures. [3]
2. (b) Find the time at which the mass is 15 000 mg. Give your answer to 3 significant figures. [2]
3. (c) At \(t=14\) a fungicide is added. From then the mass is modelled by \(M=15\,000\,e^{-0.42s}\), where s is the time in hours since the fungicide was added. Find the half-life of this decay. Give your answer to 3 significant figures. [3]

The number of species S found on an island of area x square kilometres is modelled by \(S=a+b\ln x\), where a and b are constants. An island of area 3 km² has 4.1 species per sampling plot on average and an island of area 20 km² has 8.7. [8]

1. (a) Find the value of b and the value of a. Give your answers to 3 significant figures. [4]
2. (b) Use the model to predict the number of species per plot on an island of area 45 km². Give your answer to 3 significant figures. [2]
3. (c) Use the model to find the area of an island with 12 species per plot. Give your answer to 3 significant figures. [2]