# CIE Logarithm (Additional Mathematics -2018)

$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 11, question 8)

(a) Find the value of $\D x$ for which $\D 2\lg x - \lg(5x + 60) = 1 .$ [5]

(b) Solve $\D \log_5 y = 4\log_y 5 .$ [4]

2 (CIE 2012, w, paper 11, question 3)

Given that $\D p = \log_q 32,$ express, in terms of $\D p,$

(i) $\D \log_q 4,$ [2]

(ii) $\D \log_q 16q.$ [2]

3 (CIE 2012, w, paper 12, question 4)

Given that $\D \log_a pq = 9$ and $\D \log_a p^2q = 15,$ find the value of

(i) $\D \log_a p$ and of $\D \log_a q,$ [4]

(ii) $\D \log_p a + \log_q a.$ [2]

4 (CIE 2013, s, paper 11, question 4)

(i) Given that $\D \log_4 x=\frac{1}{2},$  find the value of $\D x.$ [1]

(ii) Solve $\D 2\log_4y- \log_4 (5y-12)=\frac{1}{2}.$ [4]

5 (CIE 2013, w, paper 13, question 2)

Solve $\D 2 \lg y - \lg(5y+60)=1.$ [5]

6 (CIE 2013, w, paper 21, question 4)

Given that $\D \log_p X= 5$ and $\D \log_p Y= 2,$ find

(i) $\D \log_p X^2,$ [1]

(ii) $\D \log_p\frac{1}{X},$  [1]

(iii) $\D \log_{XY} p .$ [2]

7 (CIE 2014, s, paper 22, question 6)

(a) (i) State the value of $\D u$ for which $\D \lg u = 0.$ [1]

(ii) Hence solve $\lg |2x + 3| = 0.$ [2]

(b) Express $\D 2 \log_315- (\log_a5) (\log_3a),$ where $\D a > 1,$ as a single logarithm to base 3. [4]

8 (CIE 2014, s, paper 23, question 2)

Using the substitution $\D u= \log_3 x,$ solve, for $\D x,$ the equation $\D \log_3x -12 \log_x3= 4 .$ [5]

9 (CIE 2014, w, paper 13, question 7)

Solve the equation $\D 1+ 2 \log_5 x= \log_5(18x-9).$ [5]

10 (CIE 2014, w, paper 21, question 3)

Solve the following simultaneous equations.

$\D \begin{array}{rcl} \log_2(x+3)&=&2+\log_2y\\ \log_2(x+y)&=&3 \end{array}$ [5]

11 (CIE 2015, s, paper 21, question 1)

(a) Write $\log _{27} x$ as a logarithm to base 3 .

(b) Given that $\log _{a} y=3\left(\log _{\theta} 15-\log _{a} 3\right)+1$, express $y$ in terms of $a$.

12 (CIE $2015, \mathrm{w}$, paper 21 , question 5 )

(a) Solve the following equations to find $p$ and $q$.

\begin{aligned}8^{q-1} \times 2^{2 p+1} &=4^{7} \\9^{p-4} \times 3^{q} &=81\end{aligned}

(b) Solve the equation $\lg (3 x-2)+\lg (x+1)=2-\lg 2$.

13 (CIE $2015, \mathrm{w}$, paper 23 , question 6)

Solve the following equation.

$$\log _{2}(29 x-15)=3+\frac{2}{\log _{x} 2}$$

14 (CIE 2016 , march, paper 12, question 3)

Solve $\log _{5} \sqrt{x}+\log _{25} x=3$.

15 (CIE 2016, s, paper 21, question 3)

Do not use a calculator in this question.

(i) Find the value of $-\log _{p} p^{2}$. [1]

(ii) Find $\lg \left(\frac{1}{10^{1}}\right)$.

(iii) Show that $\frac{\lg 20-\lg 4}{\log _{5} 10}=(\lg y)^{2}$, where $y$ is a constant to be found.

(iv) Solve $\log _{r} 2 x+\log _{r} 3 x=\log _{r} 600$.

16 (CIE 2016, s, paper 22, question 10)

(a) The graph of the curve $y=p\left(4^{2 x}\right)-q\left(4^{x}\right)$ passes through the points $(0,2)$ and $(0.5,14)$. Find the value of $p$ and of $q .$

(b) The variables $x$ and $y$ are connected by the equation $y=10^{2 x}-2\left(10^{x}\right)$. Using the substitution $u=10^{x}$, or otherwise, find the exact value of $x$ when $y=24$.

(c) Solve $\log _{2}(x+1)-\log _{2} x=3$. [3]

17 (CIE $2016, \mathrm{w}$, paper 11, question 3)

By using the substitution $y=\log _{3} x$, or otherwise, find the values of $x$ for which

$3\left(\log _{3} x\right)^{2}+\log _{3} x^{5}-\log _{3} 9=0$

18 (CIE $2016, \mathrm{w}$, paper 13 , question 5)

(i) Given that $\log _{9} x y=\frac{5}{2}$, show that $\log _{3} x+\log _{3} y=5$.

(ii) Hence solve the equations $\quad \log _{9} x y=\frac{5}{2}$,

$\log _{3} x \times \log _{3} y=-6$

19 (CIE 2016, w, paper 21, question 3)

Solve the equation $2 \lg x-\lg \left(\frac{x+10}{2}\right)=1$.

20 (CIE 2017, march, paper 22 , question 6)

(a) (i) Express $\left(\sqrt[3]{-8 x^{9}}\right)\left(\sqrt[6]{x^{-3}}\right)$ in the form $a x^{b}$, where $a$ and $b$ are constants to be found.

(ii) Hence solve the equation $\left(\sqrt[3]{-8 x^{9}}\right)\left(\sqrt[6]{x^{-3}}\right)=-6250$. [2]

(b) It is given that $y=\log _{a}(a x)+2 \log _{a}(4 x-3)-1$, where $a$ is a positive integer.

(i) Explain why $x$ must be greater than $0.75$. $[1]$

(ii) Show that $y$ can be written as $\log _{a}\left(16 x^{3}-24 x^{2}+9 x\right)$. [3]

(iii) Find the value of $x$ for which $y=\log _{a}(9 x)$. [2]

21 (CIE 2017, s, paper 22, question 7)

(a) Given that $a^{7}=b$, where $a$ and $b$ are positive constants, find,

(i) $\log _{\theta} b$, $[1]$

(ii) $\log _{b} a$. [1]

(b) Solve the equation $\log _{81} y=-\frac{1}{4}$. [2]

(c) Solve the equation $\frac{32^{x^{2}-1}}{4^{x^{2}}}=16$. [3]

22 (CIE 2017, w, paper 21, question 3)

Solve the equation $\log _{5}(10 x+5)=2+\log _{5}(x-7)$

23 (CIE $2017, \mathrm{w}$, paper 22 , question 4) Solve the simultaneous equations

\begin{aligned}&\log _{2}(x+4)=2 \log _{2} y \\&\log _{2}(7 y-x)=4\end{aligned}

24 (CIE $2017, \mathrm{w}$, paper 23 , question 4) Solve the simultaneous equations

\begin{aligned}&\log _{3}(x+1)=1+\log _{3} y \\&\log _{3}(x-y)=2\end{aligned}

25 (CIE 2018, march, paper 22, question 8)

(a) Solve the following equations.

(i) $5 \mathrm{e}^{3 x+4}=14$

(ii) $\lg (2 y-7)+\lg y=2 \lg 3$

(b) Write $\frac{\log _{2} p-\log _{2} q}{\left(\log _{2} r\right)\left(\log _{r} 2\right)}$ as a single logarithm to base $2 .$

26 (CIE 2018, s, paper 11, question 6)

(a) Write $\left(\log _{2} p\right)\left(\log _{3} 2\right)+\log _{3} q$ as a single logarithm to base 3  [3]

(b) Given that $\left(\log _{a} 5\right)^{2}-4 \log _{a} 5+3=0$, find the possible values of $a$.

27 (CIE 2018, s, paper 12, question 9)

(a) (i) Solve $\lg x=3$.

(ii) Write $\lg a-2 \lg b+3$ as a single logarithm. [3]

(b) (i) Solve $x-5+\frac{6}{x}=0$.

(ii) Hence, showing all your working, find the values of $a$ such that $\log _{4} a-5+6 \log _{a} 4=0$. $[3]$

1. (a) $\D x = 60$

(b) $\D y = 25,\frac{1}{25}$

2. (i) $\D 2p/5$

(ii) $\D 1 + 4p/5$

3. (i) $\D \log_a p = 6,\log_a q = 3$

(ii) $\D 0.5$

4. (i) $\D 2$

(ii) $\D y = 4, 6$

5. $\D 60$

6. $\D 10;-5;1/7$

7. (a) $\D 1;x = -1,-2$

(b) $\log_3 45$

8. $\D 729; 1/9$

9. $\D x = 3/5; 3$

10. $\D x = 5.8; y = 2.2$

11. (a) $\frac{\log _{3} x}{3}$

(b) $y=125 a$

12. (a) $p=5, q=2(\mathrm{~b}) x=4$

13. $x=5 / 8,3$

14. 125

15. (i)-2 (ii)-n

(iv) $x=10$

16. (a) $p=5, q=3$

(b) $x=\lg 6(\mathrm{(c)}) x=1 / 7$

17. $x=3^{1 / 3}, 3^{-1}$

18. (ii) $x=729, y=1 / 3$

$x=1 / 3, y=729$

19. $x=10$

20. (a) $-2 x^{5 / 2}, 25$

(b) $\log _{a}(4 x-3)^{2}, 3 / 2$

21. (ai) 7 (ii) $1 / 7$

(b) $1 / 3($ c $) \pm \sqrt{3}$

22. $x=12$

23. $x=5, y=3$ or $x=12, y=4$

24. $x=14, y=5$

25. $-0.99,4.5, \log _{2}(p / q)$

26. $\log _{3} p q$

$1.71$

27. a(i) 1000 (ii) $\lg \frac{1000 a}{b^{2}}$

$\mathrm{~b}(\mathrm{i}) x=2, x=3$

(ii) $a=64, a=16$