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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

no calculator

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.

hard

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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