Exponents and logarithms: exam review Exam review

Use the calculator where it helps. [7]

1. (a) Solve \(10^x = 250\). Give x to 3 significant figures. [2]
2. (b) Solve \(\log_{10}(2x − 3) = 2\). Give the exact value of x. [2]
3. (c) Solve \(7 − 3\ln x = 2\). 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 \(\dfrac{(3x^4)^2}{6x^{-3}}\) in the form \(ax^p\). Write down the values of a and p. [3]
2. (b) Find the exact value of \(\left(\dfrac{3}{4}\right)^{-2}\). [2]

Do not use a calculator for this question. [7]

1. (a) Find the exact value of \(32^{-\frac{3}{5}}\). [2]
2. (b) Given that \(y=\dfrac{6}{x^2\sqrt{x}}\), write \(y^2\) in the form \(ax^k\). Write down the values of a and k. [3]
3. (c) Write \(\dfrac{x^2 + 5\sqrt{x}}{x}\) in the form \(ax^p + bx^q\). [2]

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

1. (a) Solve \(x^{\frac{2}{3}} = 49\). [2]
2. (b) Solve \(\left(\sqrt{2}\right)^x = 16^{x-3}\). 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 \(\log_{10}\left(\dfrac{10a^{3}}{\sqrt{b}}\right)\) in terms of p and q. [3]
2. (b) Write \(2 \ln 3 + \ln 20 − \ln 6\) 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 \(\left(\frac{2}{3}\right)^{-4}\). [2]
2. (b) Write \(\dfrac{(4x^2)^3}{8x^{-2}}\) in the form \(ax^p\). Write down the values of a and p. [3]

Solve each equation. [5]

1. (a) Solve \(\ln(x+3)=2\). Give x to 3 significant figures. [2]
2. (b) Solve \(2^{x+1}=8^{x-3}\). Give the exact value of x. [3]

A hard drive can store \(3.2\times10^{12}\) bytes. A photo file uses \(2.5\times10^{6}\) bytes. [5]

1. (a) Find how many photos the drive can store. Give your answer in the form \(a\times10^{k}\), where \(1\le a<10\) and \(k\in\mathbb{Z}\). [2]
2. (b) A second drive can store \(8.5\times10^{11}\) bytes. Find the total number of bytes the two drives can store. Give your answer in the form \(a\times10^{k}\). [2]
3. (c) One gigabyte is \(10^{9}\) bytes. Write down the capacity of the first drive in gigabytes. [1]

The energy E, in joules, released by an earthquake of magnitude M satisfies \(\log_{10}E=5.2+1.5M\). [5]

1. (a) Find the energy released by an earthquake of magnitude 6. 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) An earthquake of magnitude 8 and an earthquake of magnitude 5 occur. Find how many times more energy the magnitude 8 earthquake releases. Give your answer to 3 significant figures. [3]

Solve each equation. [13]

1. (a) Solve \(\log_{10}(x+4) + \log_{10}5 = 3\). Give an exact answer. [3]
2. (b) Solve \(\ln(2x+1) - \ln(x-2) = 1\). Give x to 3 significant figures. [4]
3. (c) Solve \(4^{x+1} = 7^{x-2}\). Give x to 3 significant figures. [4]
4. (d) Solve \(\log_{x}81 = \frac{4}{3}\). [2]

The number of users N (thousands) of an app, t months after it is launched, is modelled by \(N = 45\times1.12^{t}\). [8]

1. (a) Write down the value of N when t = 0, and find the value of N when t = 6. [2]
2. (b) Use your calculator to find when N reaches 200. Give t to 3 significant figures. [2]
3. (c) A second model is \(M = 120\times1.04^{t}\). 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]

A biologist studies how the metabolic rate y (watts) of an animal depends on its body mass x (kg). The masses in the study range from 0.03 kg to 5000 kg. [13]

1. (a) Write down the logarithm to base 10 of each of the values 0.03, 12, 5000. 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 = 0.7546, and for \(\log_{10}y\) against \(\log_{10}x\) it is r = 0.9996. 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 0.748 and intercept 0.553. Write the model in the form \(y=a x^k\). [3]
5. (e) Use the model to estimate y when x = 40. [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 \((0, 0.9)\) and \((5, 2.4)\). Find a and b. [3]

The mass M grams of a radioactive sample, t days after it is first measured, is modelled by \(M=250\times0.92^{t}\). [7]

1. (a) Write down the value of M when \(t=0\), and find the value of M when \(t=10\). Give your answer to 3 significant figures. [2]
2. (b) Write down the percentage of the M lost each day. [1]
3. (c) Use your calculator to find when \(M=100\). Give your answer to 3 significant figures. [2]
4. (d) The sample is safe to handle once it has less than 20 g. Find the first whole number of days after which it is safe. [2]

A culture of bacteria grows so that the number of bacteria N after t hours is modelled by \(N=N_0e^{kt}\), where \(N_0\) and k are constants. At \(t=0\) there are 1200 bacteria. At \(t=4\) there are 3100 bacteria. [8]

1. (a) Find the value of k. Give your answer to 3 significant figures. [3]
2. (b) Find the time at which there are 20 000 bacteria. Give your answer to 3 significant figures. [2]
3. (c) At \(t=12\) an antibiotic is added. From then the number of bacteria is modelled by \(N=20\,000\,e^{-0.35s}\), where s is the time in hours since the antibiotic was added. Find the half-life of this decay. Give your answer to 3 significant figures. [3]

The height H cm of a seedling d days after planting is modelled by \(H=a+b\ln d\), where a and b are constants. The seedling is 3.2 cm tall after 2 days and 9.8 cm tall after 14 days. [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 height of the seedling after 30 days. Give your answer to 3 significant figures. [2]
3. (c) Use the model to find the number of days after planting when the seedling is 15 cm tall. Give your answer to 3 significant figures. [2]