# FPM Indices and Logarithm (Chapter 1)

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1. $(2015 / \mathrm{jan} /$ paper01/q6)

(a) Solve, giving your answer to 3 significant figures, $$3^{z}-4=0$$ (3 marks)
Solve, giving your answers to 3 significant figures where appropriate,

(b) $9^{y}-13\left(3^{y}\right)+36=0$ (4 marks)

(c) $6^{x}-4\left(2^{x}\right)-3^{x}+4=0$ (5 marks)

2. (2016/jan/paper02/q1)
Find the exact solution of $$4^{(x-2)}=8^{(3 x-1)}$$ (4 marks)

3. (2017/june/paper01/q1)
Find the exact solution of the equation $$\dfrac{16}{e^{x}}-\mathrm{e}^{x}=6$$ (5 marks)

4. (2019/june/paper01/q8)

(a) Solve $5 p^{2}-9 p+4=0$ (2 marks)

(b) Hence solve $5^{2 x+1}-9\left(5^{x}\right)+4=0$
Give your answers to 3 significant figures where appropriate. (4 marks)
The curve with equation $y=5^{2 x+1}+5^{x}$ intersects the curve with equation $y=2\left(5^{x+1}\right)-4$ at two points.

(c) Find the coordinates of each of these two points. Give your answers to 3 significant figures where appropriate. (4 marks)

5. (2013/jan/paper02/q10)
Solve the equations

(a) $\log _{x} 1024=5$ (2 marks)

(b) $\log _{5}(6 y+11)=3$ (3 marks)

(c) $2 \log _{3} t+\log _{t} 9=5$ (4 marks)

6. (2013/june/paper02/q2)
Given that $2 \log _{4} x-\log _{2} y=3$

(a) show that $x=8 y$ (4 marks)

Given also that $\log _{5}(3 x+y)=4$

(b) find the value of $x$ and the value of $y$ (3 marks)

7. $(2014 / \mathrm{jan} / \mathrm{paper} 01 / \mathrm{q} 5)$

(a) Solve the equation $\log _{7}(2 x-3)=2$ (2 marks)

(b) (i) Factorise $2 x \ln 3 x-4 x-4 \ln 3 x+8$

(ii) Hence find the exact roots of the equation $2 x \ln 3 x-4 x-4 \ln 3 x+8=0$ (5 marks)

8. (2014/june/paper02/q5)
Solve the equation

(a) $\log _{x} 243=5$ (2 marks)

(b) $\log _{6}(2 y+4)=2$ (2 marks)

(c) $\log _{4} p+\log _{p} 64=4$ (5 marks)

9. (2015/june/paper02/q10)

(a) Find the value of $\log _{3} 9$ (1 mark)

Given that $\log _{9} 4=k \log _{3} 4$

(b) find the value of $k$ (2 marks)

(c) Show that (5 marks)

$$2 x \log _{3} x-3 \log _{3} x-4 x \log _{9} 4+6 \log _{9} 4=\log _{3}\left(\dfrac{x}{4}\right)^{(2 x-3)}$$

(d) Hence solve the equation $2 x \log _{3} x-3 \log _{3} x-4 x \log _{9} 4+6 \log _{9} 4=0$ (3 marks)

10. (2016/jan/paper01/q10)

Given that $2 \log _{y} x+2 \log _{x} y=5$

(a) show that $\log _{y} x=\dfrac{1}{2}$ or $\log _{y} x=2$ (5 marks)

(b) Hence, or otherwise, solve the equations

\begin{aligned}x y &=27 \\2 \log _{y} x+2 \log _{x} y &=5\end{aligned} (6 marks)

11. (2016/june/paper01/q6)
Solve

(a) $\log _{x} 1024=5$ (2 marks)

(b) $\log _{3}(7 y-3)=4$ (2 marks)

(c) $\log _{a} 25+2 \log _{a} 625=10$ (3 marks)

(d) $\log _{b} 7-2 \log _{7} b+1=0$ (5 marks)

12. $(2017 /$ june $/$ paper01/q7)

(a) Solve $\log _{a} 1024=5$ (1 marks)

(b) Solve $\log _{3}(6 c+9)=4$ (2 marks)

(c) Solve $2\left(\log _{b} 25+\log _{b} 125\right)=5$ (4 marks)

(d) Solve the equations, giving the values of $x$ and $y$ to 3 significant figures, \begin{aligned}&3 \log _{2} x+4 \log _{3} y=10 \\&\log _{2} x-2 \log _{3} y=1\end{aligned} (6 marks)

13. (2019/june/paper01/q9)

(a) Solve the equation $2 \log _{p} 9+3 \log _{3} p=8$ (6 marks)

Given that $\log _{2} 3=\log _{4} 3^{k}$

(b) find the value of $k$ (2 marks)

(c) Show that $$6 x \log _{4} x-3 x \log _{2} 3-5 \log _{4} x+10 \log _{2} 3=\log _{4}\left(\dfrac{x^{6 x-5}}{3^{6 x-20}}\right)$$ (4 marks)

14. (2019/juneR/paper01/q3)

(a) Write down the value of $\log _{3} 9$ (1 mark)

(b) Solve the equation $\log _{3} 9 t=\log _{9}\left(\dfrac{12}{t}\right)^{2}+2 \quad$ where $t>0$

Give your answer in the form $a \sqrt{b}$ where $a$ and $b$ are prime numbers. (6 marks)

15. (2011/june/paper01/q7)

(a) Solve

$$5 p^{2}-11 p+2=0$$ (2 marks)

(b) Hence solve $5\left(3^{2 x}\right)-11\left(3^{x}\right)+2=0$ giving your answers to 3 significant figures. (4 marks)

The curve with equation $y=5\left(3^{2 x}\right)-6\left(3^{x}\right)$ intersects the curve with equation $y=5\left(3^{x}\right)-2$ at two points.

(c) Find the coordinates of each of these two points, giving your answers to 3 significant figures where appropriate. (4 marks)

16. (2012/june/paper02/q1)

Solve the equation

$$5^{x+1}=120$$

giving your answer to 3 significant figures. (4 marks)

17. $(2017 / \mathrm{jan} /$ paper02/q7)

(a) Given that $k$ is a constant such that $\dfrac{27^{(x+2)}-3^{(3 x+5)}}{3^{x} \times 9^{(x+2)}}=k$ find the value of $k$ (5 marks)

(b) Find the exact roots of the equation $2 \log _{2} y+3 \log _{y} 2=7$ (6 marks)

18. $(2018 / \mathrm{jan} /$ paper02/q7)

(i) Solve the equation $\dfrac{\left(8^{x}\right)^{x}}{32^{x}}=4$ (4 marks)

(ii) Solve the equation $\log _{x} 64+3 \log _{4} x-\log _{x} 4=5$ (7 marks)

19. (2018/june/ paper02/q4)

(a) Find the exact value of the root of the equation $\mathrm{e}^{2 n}=8$

Give your answer in the form $\ln a$, where $a$ is an integer. (2 marks)

The curve $C_{1}$ has equation $y=2 \mathrm{e}^{3x}$ and the curve $C_{2}$ has equation $y=\left(\mathrm{e}^{{3x}}-4\right)^{2}$

The curves $C_{1}$ and $C_{2}$ intersect at the points $P$ and $Q$.

(b) Use algebra to find the exact coordinates of the points $P$ and $Q$. (5 marks)

(c) Find, to 3 decimal places, the length of $P Q$. (2 marks)

20. (2011/june/paper01/q2)

(a) Given that $\log _{a} x=\dfrac{\log _{b} x}{\log _{b} a}$ show that $\log _{a} b=\dfrac{1}{\log _{b} a}$ (2 marks)

(b) Hence solve the equation $$\log _{x} 8-6 \log _{g} x=1 \quad x \in \mathbb{Z}^{+}$$ (5 marks)

### Answer

1.(a) $z=1,26$ (b) $y=1.26,2$ (c) $x=1,26,0$

2. $x=-\dfrac{1}{7}$

3. $x=\ln 2$

4. (a) $p=\dfrac{4}{5}$ or $p=1$ (b) $x=0$ or $-0.139$ (c) $(0,6),(-0.139,4)$

5.(a) $x=4$ (b) $y=19$ (c) $t=\sqrt{3}$

6.(a) show (b) $x=200, y=25$

7.(a) $x=26$ (b) $(i)(2 x-4)(\ln 3 x-2)$ (ii) $x=2, \dfrac{1}{3} e^{2}$

8.(a) $x=3$ (b) $y=16$ (c) $p=4,64$

9.(a) 2 (b) $k=\dfrac{1}{2}$ (c) Show (d) $x=\dfrac{3}{2}, 4$

10.(a) Show (b) $x=3, y=9$ or $x=9, y=3$

11.(a) $\quad x=4$ (b) $y=12$ (c) $a=5$ (d) $b=7$

12. (a) $a=4$ (b) $c=12$ (c) $b=25$ (d) $x=5.28, y=2 \cdot 16$

13. (a) $p=\sqrt[3]{9}$ or 9 (b) $k=2$ (c) Show

14.(a) 2 (b) $t=2\sqrt 3$

15.(a) $\quad P=\dfrac{1}{5}, 2$ (b) $x=-1.46,0.631$ (c) $(-1.46,-1),(0.631,8)$

16. $x=1.97$

17. (a) $k=6$ (b) $y=8$ or $\sqrt{2}$

18.(i) $x=2,-\dfrac{1}{3}$ (ii) $x=4,4^{\frac{2}{3}}$

19.(a) $x=\ln 2$ (b) $P=(\ln 2,16),Q=(\dfrac{1}{3}\ln 2,4)$ (c) 12.009

20.(a) Show (b) $x=2$