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. [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]

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]

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]