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IB Mathematics AA SL - Questionbank

Exponents & Logs

Exponent & Log Laws, Solving Exponential & Logarithmic Equations…

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Question 1

no calculator

easy

[Maximum mark: 7]

Find the value of each of the following, giving your answer as an integer.

  1. log10100\log_{10} 100. [2]

  2. log1050+log102\log_{10} 50 + \log_{10} 2. [2]

  3. log104log1040\log_{10} 4 - \log_{10} 40. [3]

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Question 2

no calculator

easy

[Maximum mark: 6]

Find the value of each of the following, giving your answer as an integer.

  1. log66\log_6 6. [2]

  2. log69+log64\log_6 9 + \log_6 4. [2]

  3. log672log62\log_6 72 - \log_6 2. [2]

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Question 3

no calculator

easy

[Maximum mark: 6]

Let log2a=p\log_2 a = p, log2b=q\log_2 b = q, log2c=r\log_2 c = r. Write down the following expressions in terms of pp, qq and rr.

  1. log2(abc)\log_2\Big(\dfrac{ab}{c}\Big) [2]

  2. log2(a2cb3)\log_2\Big(\dfrac{a^2c}{b^3}\Big) [2]

  3. logab\log_a b [2]

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Question 4

no calculator

easy

[Maximum mark: 7]

Let p=ln2p=\ln 2 and q=ln6q = \ln 6. Write down the following expressions in terms of pp and qq.

  1. ln12\ln 12 [2]

  2. ln3\ln 3 [2]

  3. ln48\ln 48 [3]

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Question 5

no calculator

easy

[Maximum mark: 6]

Let a=log5ba = \log_5b, where b>0b > 0. Write down each of the following expressions
in terms of aa.

  1. log5b4\log_5b^4 [2]

  2. log5(25b)\log_5 (25b) [2]

  3. log25b\log_{25}b [2]

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Question 6

no calculator

medium

[Maximum mark: 5]

Consider a=log6364×log6263×log6162××log23a = \log_{63}64\times\log_{62}63\times\log_{61}62\times\dots\times\log_{2}3. Given that aZa\in\mathbb{Z}, find the value of aa.

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Question 7

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medium

[Maximum mark: 6]

  1. Write the expression 4ln2ln84\ln 2 - \ln 8 in the form of lnk\ln k, where kZk \in \mathbb{Z}. [3]

  2. Hence, or otherwise, solve 4ln2ln8=ln(2x)4\ln 2 - \ln 8 = -\ln (2x). [3]

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Question 8

no calculator

medium

[Maximum mark: 5]

Solve the equation log5xlog54=2+log53\log_5 x - \log_5 4 = 2 + \log_5 3 for xx.

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Question 9

calculator

medium

[Maximum mark: 6]

Given that loga2=5\log_a 2 = 5.

  1. Find the exact value of loga32\log_a 32. [2]

  2. Find the exact value of loga2\log_{\sqrt{a}} 2. [2]

  3. Find the value of aa, giving your answer correct to 33 significant figures. [2]

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Question 10

no calculator

medium

[Maximum mark: 5]

Find the values of xx when 27x+2=(19)2x+427^{x+2} = \left(\dfrac{1}{9}\right)^{2x+4}.

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Question 11

no calculator

medium

[Maximum mark: 5]

Solve the equation log3xlog35=1+log34\log_3 x - \log_3 5 = 1 + \log_3 4 for xx.

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Question 12

no calculator

medium

[Maximum mark: 6]

  1. Write down the value of

    1. log28\log_2 8;

    2. log5(125)\log_5\Big(\dfrac{1}{25}\Big);

    3. log93\log_9 3. [3]

  2. Hence solve log28+log5(125)+log93=log16x\log_2 8 + \log_5\Big(\dfrac{1}{25}\Big) + \log_9 3 = \log_{16} x.[3]

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Question 13

no calculator

medium

[Maximum mark: 6]

  1. Write down the value of

    1. log381\log_3 81;

    2. log2(18)\log_2\Big(\dfrac{1}{8}\Big);

    3. log255\log_{25} 5. [3]

  2. Hence solve log381+log2(18)+log255=log9x\log_3 81 + \log_2\Big(\dfrac{1}{8}\Big) + \log_{25} 5 = \log_{9} x.[3]

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Question 14

calculator

medium

[Maximum mark: 6]

Given that logb3=10\log_b 3 = 10.

  1. Find the exact value of logb81\log_b 81. [2]

  2. Find the exact value of logb23\log_{b^2} 3. [2]

  3. Find the value of bb, giving your answer correct to 33 significant figures. [2]

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Question 15

no calculator

medium

[Maximum mark: 5]

Find the values of xx when 25x22x=(1125)4x+225^{x^2-2x} = \left(\dfrac{1}{125}\right)^{4x+2}.

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Question 16

no calculator

medium

[Maximum mark: 5]

Solve the equation log3(x24x+4)=1+log3(x2)\log_3(x^2-4x+4) = 1 + \log_3(x-2).

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Question 17

no calculator

medium

[Maximum mark: 5]

Solve the equation log2(x22x+1)=1+log2(x1)\log_2(x^2-2x+1) = 1 + \log_2(x-1).

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Question 18

no calculator

medium

[Maximum mark: 5]

Solve log6(x)+log6(x5)=2\log_6(x) + \log_6(x-5) = 2, for x>5x > 5.

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Question 19

no calculator

medium

[Maximum mark: 6]

Find the value of

  1. log575log53\log_5 75 - \log_5 3; [2]

  2. 25log5825^{\log_5 8}. [4]

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Question 20

no calculator

medium

[Maximum mark: 6]

Find the value of

  1. log798log72\log_7 98 - \log_7 2; [2]

  2. 49log7649^{\log_7 6}. [4]

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Question 21

no calculator

medium

[Maximum mark: 15]

The equation e2x12ex=32e^{2x} - 12e^x = -32 has two solutions, x1x_1 and x2x_2.

  1. Find the value of x1x_1 and the value of x2x_2.[5]

A second equation, 2(log9x)(log3x)6log9x2log3x=62\left(\log_{\,9}x\right)\left(\log_{\,3}x\right) - 6\log_{\,9}x - 2\log_{\,3}x = -6, also has two solutions, x3x_3 and x4x_4.

    1. Show that this second equation can be expressed as

      (log3x)25log3x+6=0\begin{align*} \left(\log_{\,3}x\right)^2 -5\log_{\,3}x + 6 = 0 \end{align*}
    2. Hence find the value of x3x_3 and the value of x4x_4. [7]

  1. Given that x1+x2=a(x3+x4)x_1 + x_2 = a(x_3 + x_4), find the value of aa. Give your answer in the form blncb\ln c, where b,cRb,c \in \mathbb{R}.[3]

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Question 22

no calculator

medium

[Maximum mark: 18]

The first three terms of an infinite sequence, in order, are

2lnx,qlnx,lnx where  x>0.2\ln x,\,\, q\ln x,\,\, \ln \sqrt{x}\,\,\, \text{ where $\ x > 0$.}

First consider the case in which the series is geometric.

    1. Find the possible values of qq.

    2. Hence or otherwise, show that the series is convergent. [3]

  1. Given that q>0q>0 and S=8ln3S_\infty=8\ln{3}, find the value of xx. [3]

Now suppose that the series is arithmetic.

    1. Show that q=54q=\dfrac{5}{4}.

    2. Write down the common difference in the form mlnxm\ln x, where mQm \in \mathbb{Q}. [4]

  1. Given that the sum of the first nn terms of the sequence is lnx5\ln \sqrt{x^5}, find the value of nn. [8]

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Question 23

no calculator

hard

[Maximum mark: 5]

Solve the equation 9x+23x+1=19^x + 2\cdot3^{x+1} = 1.

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Question 24

no calculator

hard

[Maximum mark: 7]

Solve the simultaneous equations:

1+2log5x=log57ylog7(6x1)=1+log7y\begin{aligned} 1 + 2\log_5 x &= \log_5 7y \\[6pt] \log_7 (6x-1) &= 1 + \log_7 y\end{aligned}

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Question 25

no calculator

hard

[Maximum mark: 14]

The first two terms of an infinite geometric sequence, in order, are

3log3x,2log3x,where x>0.3\log_3x,\,\, 2\log_3x,\,\, \text{where $x > 0$.}
  1. Find the common ratio, rr. [2]

  2. Show that the sum of the infinite sequence is 9log3x9\log_3 x. [3]

The first three terms of an arithmetic sequence, in order, are

log3x,log3x3,log3x9,where x>0.\log_3x,\,\, \log_3 \dfrac{x}{3},\,\, \log_3\dfrac{x}{9},\,\, \text{where $x > 0$.}
  1. Find the common difference dd, giving your answer as an integer. [3]

Let S6S_6 be the sum of the first 66 terms of the arithmetic sequence.

  1. Show that S6=6log3x15S_6 = 6\log_3 x - 15. [3]

  2. Given that S6S_6 is equal to one third of the sum of the infinite geometric
    sequence, find xx, giving your answer in the form apa^p where a,pZa,p \in \mathbb{Z}. [3]

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Question 26

no calculator

hard

[Maximum mark: 7]

Consider f(x)=logk(8x2x2)f(x) = \log_k(8x-2x^2), for 0<x<40 < x < 4, where k>0k > 0.

The equation f(x)=3f(x) = 3 has exactly one solution. Find the value of kk.

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Question 27

no calculator

hard

[Maximum mark: 6]

Solve log3(sinx)log3(cosx)=1\log_{\sqrt{3}}(\sin x) - \log_{\sqrt{3}}(\cos x) = 1, for 0<x<π20 < x < \dfrac{\pi}{2}.

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Question 28

no calculator

hard

[Maximum mark: 5]

Solve the equation 154a=81a+215^{4a} = 81^{a+2} for aa. Express your answer in terms of ln3\ln 3 and ln5\ln 5.

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Question 29

no calculator

hard

[Maximum mark: 5]

Find the integer values of aa and bb for which

a+blog47+60log814=0.\begin{aligned} a + b\log_4 7 + 60\log_{8} 14 &= 0.\end{aligned}

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Question 30

calculator

hard

[Maximum mark: 5]

Solve the equation 146x=64x+314^{6x} = 64^{x+3} for xx. Express your answer in terms of ln2\ln 2 and ln7\ln 7.

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Question 31

no calculator

hard

[Maximum mark: 8]

The first three terms of a geometric sequence are lnx9\ln x^9, lnx3\ln x^3, lnx\ln x, for x>0x > 0.

  1. Find the common ratio. [3]

  2. Solve k=133klnx=27\displaystyle \sum_{k=1}^\infty 3^{3-k}\ln x = 27. [5]

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Question 32

no calculator

hard

[Maximum mark: 8]

  1. Show that log23cos2x=log4(3cos2x)\log_2 \sqrt{3-\cos 2x} = \log_4 (3-\cos 2x). [3]

  2. Hence, or otherwise, solve log4(3sinx)+14=log23cos2x\log_4 (3\sin x) \hspace{0.15em}+\hspace{0.15em} \dfrac{1}{4} = \log_2 \sqrt{3-\cos 2x}, for 0<x<π20 \hspace{-0.05em}<\hspace{-0.05em} x \hspace{-0.05em}<\hspace{-0.05em}\dfrac{\pi}{2}.[5]

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Question 33

no calculator

hard

[Maximum mark: 8]

  1. Show that log16(cos2x+7)=log4cos2x+7\log_{16}(\cos 2x + 7) = \log_4 \sqrt{\cos 2x + 7}. [3]

  2. Hence, or otherwise, solve log4(10cosx)=log16(cos2x+7)\log_4(\sqrt{10}\cos x) = \log_{16}(\cos 2x + 7), for 0<x<π20 < x < \dfrac{\pi}{2}.[5]

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Frequently Asked Questions

The IB Math Analysis and Approaches (AA) SL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. The bank of exam style questions are accompanied by high quality step-by-step markschemes and video tutorials, taught by experienced IB Mathematics teachers. The IB Mathematics AA SL Question bank is the perfect exam revision resource for IB students looking to practice IB Math exam style questions in a particular topic or concept in their AA Standard Level course.

The AA SL Questionbank is designed to help IB students practice AA SL exam style questions in a specific topic or concept. Therefore, a good place to start is by identifying a concept that you would like to practice and improve in and go to that area of the AA SL Question bank. For example, if you want to practice AA SL exam style questions that have Exponents & Logarithms in them, you can go to AA SL Topic 1 (Number & Algebra) and go to the Exponents & Logarithms area of the question bank. On this page there is a carefully designed set of IB Math AA SL exam style questions, progressing in order of difficulty from easiest to hardest. If you’re just getting started with your revision, you could start at the top of the page with Question 1, or if you already have some confidence, you could start at the medium difficulty questions and progress down.

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With an extensive and growing library of full length IB Math Analysis and Approaches (AA) SL exam style questions in the AA SL Question bank, finishing all of the questions would be a fantastic effort, and you will be in a great position for your final exams. If you were able to complete all the questions in the AA SL Question bank, then a popular option would be to go to the AA SL Practice Exams section on Revision Village and test yourself with the Mock Exam Papers, to simulate the length and difficulty of an actual IB Mathematics AA SL exam.