CIE Logarithm and Indices (-2020)

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1. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 21 / \mathrm{q} 3)$ 
(a) Solve $\mathrm{e}^{2 x+1}=3 \mathrm{c}^{4-3 x}$ 
(b) Solve $\lg (y-6)+\lg (y+15)=2$ 

 2. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 21 / \mathrm{q} 2)$ 
(a) Solve $3^{\left(\frac{x}{2}-1\right)}=10$ 
(b) Solve $2 e^{1-2 y}=3 e^{3 y+2}$

 3. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 22 / \mathrm{q} 4)$ 
Solve (i) $2^{3 x-1}=6$ (ii) $\log _{3}(y+14)=1+\frac{2}{\log _{y} 3}$

4. (CIE $0606 / 2019 / \mathrm{w} / 11 / \mathrm{q} 3)$ 
Given that $7^{x} \times 49^{y}=1$ and $5^{5 x} \times 125^{\frac{2 v}{3}}=\frac{1}{25}$, calculate the value of $x$ and of $y$

5. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 12 / \mathrm{q} 4)$ 
(a) Given that $\frac{\left(p r^{2}\right)^{\frac{3}{2}} \sqrt{q r}}{q^{2}\left(p r^{2}\right)^{-1}}$ can be written in the form $p^{n} q^{b} r^{c}$, find the value of each of the constants $a$, $b$ and $c$. 
(b) Solve $$\begin{gathered} 3 x^{\frac{1}{2}}-y^{-\frac{1}{2}}=4 \\ 4 x^{\frac{1}{2}}+3 y^{-\frac{1}{2}}=14 \end{gathered}$$ 

 6. (CIE 0606/2019/w/12/q6) 
(a) Write $\frac{\sqrt{p}\left(\frac{q p}{r}\right)^{2}}{p^{-1} \sqrt[3]{q r}}$ in the form $p^{a} q^{b} r^{c}$, where $a, b$ and $c$ are constants. 
(b) Solve $\log _{7} x+2 \log _{x} 7=3$ 

 7. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 13 / \mathrm{q} 2)$ 
Given that $\frac{\sqrt{p}(q r)^{-2}}{p^{2} q^{\frac{1}{3}} r}=\frac{1}{p^{a} q^{b} r^{c}}$, find the value of each of the constants $a, b$ and $c_{\text {. }}$ 

 8. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 11 / \mathrm{q} 3)$ 
(a) Write $\frac{\sqrt{p}\left(q r^{2}\right)^{\frac{1}{3}}}{\left(q^{3} p\right)^{-1} r^{3}}$ in the form $p^{a} q^{b} r^{c}$, where $a, b$ and $c$ are constants. 
(b) Solve $6 x^{\frac{2}{3}}-5 x^{\frac{1}{3}}+1=0$ 

 9. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 21 / \mathrm{q} 7)$ 
(a) Solve the simultaneous equations $$\begin{aligned} 10^{x+2 y} &=5 \\ 10^{3 x+4 y} &=50 \end{aligned}$$ giving $x$ and $y$ in exact simplified form. 
(b) Solve $2 x^{\frac{2}{3}}-x^{\frac{1}{3}}-10=0$ 

 10. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 22 / \mathrm{q} 2)$ 
Find the value of $x$ such that $\frac{4^{x+1}}{2^{x-1}}=32^{\frac{x}{3}} \times 8^{\frac{1}{3}}$ 

 11. $(\mathrm{CIE} 0606 / 2020 / \mathrm{m} / 22 / \mathrm{q} 3)$ 
The position vectors of three points, $A, B$ and $C$, relative to an origin $O$, are $\left(\begin{array}{l}-5 \\ -7\end{array}\right),\left(\begin{array}{c}10 \\ -4\end{array}\right)$ and $\left(\begin{array}{l}x \\ y\end{array}\right)$ respectively. Given that $\overrightarrow{A C}=4 \overrightarrow{B C}$, find the unit vector in the direction of $\overrightarrow{O C}$ 

 12. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 22 / \mathrm{q} 9)$ 

(a) Solve the equation $\frac{9^{5 x}}{27^{x-2}}=243$ (b) $\quad \log _{a} \sqrt{b}-\frac{1}{2}=\log _{b} a$, where $a>0$ and $b>0$. Solve this equation for $b$, giving your answers in terms of $a$. $[5]$ 

 13. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 11 / \mathrm{q} 7)$ 
(a) Express $2+3 \lg x-\lg y$ as a single logarithm to base 10 . 
(b) (i) Solve $6 x+7-\frac{3}{x}=0$. (ii) Hence, given that $6 \log _{a} 3+7-3 \log _{5} a=0$, find the possible values of $a$. [4] 

 14. (CIE $0606 / 2018 / \mathrm{w} / 13 / \mathrm{q} 5)$ 
(i) Show that $\log _{9} 4=\log _{3} 2$ 
(ii) Hence solve $\log _{9} 4+\log _{3} x=3$ 

 15. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 11 / \mathrm{q} 7)$ 
(a) Solve $\log _{3} x+\log _{9} x=12$. 
(b) Solve $\log _{4}\left(3 y^{2}-10\right)=2 \log _{4}(y-1)+\frac{1}{2}$ 

 16. $(\mathrm{CIE} 0606 / 2019 / \mathrm{m} / 12 / \mathrm{q} 5)$ 
It is given that $\log _{4} x=p$. Giving your answer in its simplest form, find, in terms of $p$, (i) $\log _{4}(16 x)$, (ii) $\log _{4}\left(\frac{x^{\top}}{256}\right)$ Using your answers to parts (i) and (ii), (iii) solve $\log _{4}(16 x)-\log _{4}\left(\frac{x^{\top}}{256}\right)=5$, giving your answer correct to 2 decimal places. 

 17. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 13 / \mathrm{q} 8)$ 
(a) Given that $\log _{\theta} x=p$ and $\log _{\alpha} y=q$, find, in terms of $p$ and $q$, 
(i) $\log _{a} a x y^{2}$ 
(ii) $\log _{0}\left(\frac{x^{3}}{a y}\right)$, 
(iii) $\log _{x} a+\log _{y} a$. $[1]$ (b) Using the substitution $m=3^{x}$, or otherwise, solve $3^{x}-3^{1+2 x}+4=0$. [3] 

 18. (CIE 0606/2019/s/22/q7) 
(a) Solve $\lg \left(x^{2}-3\right)=0$. [2] 

19. (CIE 0606/2019/s/22/q7(b)) 
(i) Show that, for $a>0, \frac{\ln a^{\sin (2 x+5)}+\ln \left(\frac{1}{a}\right)}{\ln a}$ may be written as $\sin (2 x+5)+k$, where $k$ is an integer. 
(ii) Hence find $\displaystyle\int \frac{\ln a^{\sin (2 x+5)}+\ln \left(\frac{1}{a}\right)}{\ln a} \mathrm{~d} x$ $[3]$ 


 20. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 21 / \mathrm{q} 3)$ 
Write $3 \lg x+2-\lg y$ as a single logarithm. 

 21. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 22 / \mathrm{q} 4)$ 
Solve the simultaneous equations. $$\begin{aligned} \log _{3}(x+y) &=2 \\ 2 \log _{3}(x+1) &=\log _{3}(y+2) \end{aligned}$$ 

 22. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 23 / \mathrm{q} 8)$ 
DO NOT USE A CALCULATOR IN THIS QUESTION. 
$$\begin{aligned} &\log _{2}(y+1)=3-2 \log _{2} x \\ &\log _{2}(x+2)=2+\log _{2} y \end{aligned}$$ 
(a) Show that $x^{3}+6 x^{2}-32=0$. 
(b) Find the roots of $x^{3}+6 x^{2}-32=0$. (c) Give a reason why only one root is a valid solution of the logarithmic equations. Find the value of $y$ corresponding to this root. 

 23. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 11 / \mathrm{q} 7)$ 
It is given that $\mathrm{f}(x)=5 \ln (2 x+3)$ for $x>-\frac{3}{2}$ 
(a) Write down the range of $f$.[1] 
(b) Find $\mathrm{f}^{-1}$ and state its domain. [3] 
(c) On the axes below, sketch the graph of $y=\mathrm{f}(x)$ and the graph of $y=\mathrm{f}^{-1}(x)$. Label each curve and state the intercepts on the coordinate axes. 

Answer
1. (a) $x=\frac{3+\ln 3}{5}$
(b) $y=10$
2. (a) $x=6.19$
(b) $y=-0.281$
3. (i) $x=1.19$
(ii) $y=\frac{7}{3}$
4. $\quad x=-\frac{1}{2}, y=\frac{1}{4}$
5. (a) $p^{\frac{5}{2}} q^{-\frac{3}{2}} r^{\frac{11}{2}}$
(b) $x=4, y=\frac{1}{4}$
6. (a) $p^{\frac{7}{2}} q^{\frac{5}{3}} r^{-\frac{7}{3}}$
(b) $x=7, x=49$
7. $\quad a=\frac{3}{2}, b=\frac{7}{3}, c=3$
8. $(a) p^{\frac{3}{2}} q^{\frac{10}{3}} r^{-\frac{7}{3}}$
(b) $x=\frac{1}{27}, \frac{1}{8}$
9. (a) $x=\lg 2, y=\frac{1}{2} \lg \frac{5}{2}$
(b) $x=-8, \frac{125}{8}$
10. $x=3$
11. $x=\log _{3} 4$
12. (a) $x=-\frac{1}{7}$
(b) $b=a^{2}$
13. (a) $\lg \frac{100 x^{3}}{y}$
(b) (i) $x=-\frac{3}{2}, \frac{1}{3}$
(ii) $0.481$
14. (i) Show (ii) $x=\frac{27}{2}$
15. (a) $x=6561$ (b) $y=2$
16. (i) $2+p$
(ii) $7 p-4$
(iii) $x=1.26$
17. (a)(i) $1+2 p+q$ (ii) $3 p-q-1$
(iii) $\frac{1}{p}+\frac{1}{q}$
(b) $x=0.262$
18. (a) $x=\pm 2$
19(i) $\sin (2 x+5)-1$
(ii) $-\frac{1}{2} \cos (2 x+5)-x(+c)$
20. $\lg \left(\frac{100 x^{3}}{y}\right)$
21. $\quad x=2, y=7$
22. (a) Proof
(b) $x=2,-4,-4$ (c) $y=1$
23. (a) $f \in \mathbb{R}$
(b) $f^{-1}(x)=\frac{e^{\frac{2}{5}}-3}{2}$, Domain: $x \in \mathbb{R}$

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