Exponents and logarithms: exam review Exam review Worked solutions

Use the calculator where it helps. [7]

1. (a) Solve \(10^x = 250\). Give x to 3 significant figures. [2]

   • M1 for x = log10 250
   • A1 2.40

2. (b) Solve \(\log_{10}(2x − 3) = 2\). Give the exact value of x. [2]

   • M1 for 2x − 3 = 10^2 = 100
   • A1 x = \frac{103}{2} (51.5)

3. (c) Solve \(7 − 3\ln x = 2\). Give the exact value of x, and x to 3 significant figures. [3]

   • M1 for isolating the log: ln x = \frac{5}{3}
   • M1 for x = e^(\frac{5}{3})
   • A1 5.29

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]

   • M1 for (3x^4)^2 = 9x^8
   • M1 for dividing by x^(−3), so the indices are added: 8 + 3
   • A1 a = 3/2, p = 11

2. (b) Find the exact value of \(\left(\dfrac{3}{4}\right)^{-2}\). [2]

   • M1 for turning the fraction over: (4/3)^2
   • A1 16/9

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

1. (a) Find the exact value of \(\left(\frac{2}{3}\right)^{-4}\). [2]

   • M1 for flipping the fraction: \(\left(\frac{3}{2}\right)^{4}\)
   • A1 \(\frac{81}{16}\)

2. (b) Write \(\dfrac{(4x^2)^3}{8x^{-2}}\) in the form \(ax^p\). Write down the values of a and p. [3]

   • M1 for \((4x^2)^3=64x^{6}\)
   • M1 for dividing the numbers and subtracting the powers
   • A1 \(a = 8\), p = 8

Solve each equation. [5]

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

   • M1 for \(x+3=e^2\)
   • A1 \(x = e^2-3 = 4.39\)

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

   • M1 for writing \(8^{x-3}=2^{3(x-3)}\)
   • M1 for \(x+1=3x-9\)
   • A1 x = 5

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]

   • M1 for \(\dfrac{3.2\times10^{12}}{2.5\times10^{6}}\) (\(1.28\times10^{6}\), the a values divided and the indices subtracted)
   • A1 \(1.28\times10^{6}\)

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]

   • M1 for writing both numbers with the same power of 10, for example \(3.2\times10^{12}+0.85\times10^{12}\)
   • A1 \(4.05\times10^{12}\)

3. (c) One gigabyte is \(10^{9}\) bytes. Write down the capacity of the first drive in gigabytes. [1]

   • A1 3200 (\(3.2\times10^{12}\div10^{9}=3.2\times10^{3}\))

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]

   • M1 for \(\log_{10}E=5.2+1.5\times6=14.2\), so \(E=10^{14.2}\)
   • A1 \(1.58\times10^{14}\)

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]

   • M1 for \(\log_{10}E_8-\log_{10}E_5=1.5\times(8-5)=4.5\) (the 5.2 cancels)
   • M1 for \(\log_{10}\dfrac{E_8}{E_5}=4.5\), so \(\dfrac{E_8}{E_5}=10^{4.5}\)
   • A1 3.16e+04

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]

   • A1 N = 45
   • A1 88.8 (45 × 1.12^6)

2. (b) Use your calculator to find when N reaches 200. Give t to 3 significant figures. [2]

   • M1 for solving 45 × 1.12^t = 200
   • A1 13.2

3. (c) A second model is \(M = 120\times1.04^{t}\). Find the time when M = N. Give t to 3 significant figures. [2]

   • M1 for solving 45 × 1.12^t = 120 × 1.04^t
   • A1 13.2

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

   • M1 for solving 1.12^t = 10
   • A1 20.3

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]

   • A1 250
   • A1 109 (\(250\times0.92^{10}\))

2. (b) Write down the percentage of the M lost each day. [1]

   • A1 8% (0.92 means a decrease of 8%)

3. (c) Use your calculator to find when \(M=100\). Give your answer to 3 significant figures. [2]

   • M1 for solving \(250\times0.92^{t}=100\)
   • A1 t = 11.0

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]

   • M1 for solving \(250\times0.92^{t}=20\) (t = 30.3)
   • A1 31 (the first whole number above 30.3)