# Multiple Choice : Logarithms

တက္ကသိုလ်ဝင်တန်း မေးခွန်းမှာ MCQ Format မပါတော့ပေမယ့် Multiple Choice Question ဆိုတာ ကျောင်းသားရဲ့ ဘာသာရပ်ဆိုင်ရာ နားလည်တတ်သိမှု၊ ဖြတ်ထိုးဉာဏ်၊ ဆင်ခြင်နိုင်စွမ်း စတာတွေကို စစ်ဆေးတာဖြစ်လို့ လေ့ကျင့်ထားသင့်ပါတယ်။ နိုင်ငံတကာ တက္ကသိုလ်ဝင် စာမေးပွဲများမှာလည်း MCQ ကိုပဲ ဦးစားပေး မေးလေ့ရှိတာမို့ နိုင်ငံရပ်ခြား ကျောင်းတက်ဖို့ စာမေးပွဲဖြေဆိုမည့်သူများ အတွက်လည်း အသုံးဝင်ပါလိမ့်မယ်။

# Definition: Logarithm

Let $N$ and $b$ be positive real numbers, with $b \neq 1$. Then the logarithm of $N$ (with respect) to the base $b$ is the exponent by which $b$ must be raised to yield $N$, and is denoted by $\log _{b} N$

# Rules of Logarithms

$\begin{array}{ll} \text{L}1. & N=b^{\log _{b} N}\\\\ \text{L}2. & x=\log _{b} b^{x}\\\\ \text{L}3. & \log _{b} b=1\\\\ \text{L}4. & \log _{b} 1=0\\\\ \text{L}5. & \log _{b}(M N)=\log _{b} M+\log _{b} N\\\\ \text{L}6. & \log _{b} N^{p}=p \log _{b} N\\\\ \text{L}7. & \log _{b}\left(\displaystyle\frac{M}{N}\right)=\log _{b} M-\log _{b} N\\\\ \text{L}8. & \log _{a} N=\displaystyle\frac{\log _{b} N}{\log _{b} N}\\\\ \text{L}9. & \log _{a} N=\displaystyle\frac{1}{\log _{N} a}\\\\ \text{L}10. & \log _{a^{p}} N=\displaystyle\frac{1}{p} \log _{a} N\\\\ \text{L}11. & a^{\log _{k} b}=b^{\log _{k} a} \end{array}$

# Common Logarithm

The logarithm of $N$ to the base $10\left(\log _{10} N\right)$ is said to be a common logarithm, and is usually written as $\log N$ (omitting the base). where $n$ is called the characteristic and $\log a$ is called the mantissa of $\log N$.

$\begin{array}{|l|} \hline\log _{10} N=\log N\\ \hline \end{array}$

If $\quad N=a \times 10^{n}$,

then $\quad \log N=\log \left(a \times 10^{n}\right)=\log 10^{n}+\log a=n+\log a$ where $n$ is called the characteristic and $\log a$ is called the mantissa of $\log N$.

Note that $n$ is an integer and $1 \leq a<10$

# Euler's Number

As a positive integer $n$ become very large, the value of $\left(1+\displaystyle\frac{1}{n}\right)^{n}$ approaches an irrational number, which is denoted by $e$.

# Natural Logarithm

The logarithm of $N$ to the base $e$ is called a natural logarithm, and is denoted by $\ln N$.

$\begin{array}{|l|} \hline\log _{e} N=\ln N\\ \hline \end{array}$

# MCQ Test

1. If $\log _{10} x=3$ then $x=$
A. $500$
B. $\displaystyle\frac{10}{3}$
C. $700$
D. $1000$
2. If $\log _{7} x=2$, then $x=$
A. $14$
B. $49$
C. $128$
D. $64$
3. The characteristic of log 19 is
A. $0$
B. $10$
C. $2$
D. $1$
4. The characteristic of $\log 3.216$ is
A. $0$
B. $4$
C. $3$
D. $10$
5. Common logarithm has the base
A. $2$
B. $e$
C. $\pi$
D. $10$
6. In scientific notation $0.00416$ is written as
A. $0.0416 \times 10^{-1}$
B. $0.416 \times 10^{-2}$
C. $4.16 \times 10^{-3}$
D. $41.6 \times 10^{-4}$
7. In decimal form $2.35 \times 10^{-2}$ is written as
A. $2.35$
B. $0.0235$
C. $0.00235$
D. $0.000235$
8. $\log 5+\log 8-\log 3=$
A. $5 \log \displaystyle\frac{8}{3}$
B. $3 \log 40$
C. $\log \displaystyle\frac{40}{3}$
D. $3 \log \displaystyle\frac{5}{8}$
9. $\log 50$ can be written as
A. $\log 2+2 \log 5$
B. $\log 2+\log 15$
C. $\log 2+5\log 2$
D. $\log 2+\log 5$
10. 3 is the characteristic in the logarithm of the number
A. $879.2$
B. $87.92$
C. $8.792$
D. $8792$
11. If $\log _{2} 8=x$ then $x=$
A. $2^{8}$
B. $64$
C. $3^{2}$
D. $3$
12. $5^{4}=625$ is written in the logarithmic form as
A. $\log 5=625$
B. $\log _{5} 4=625$
C. $\log _{5} 625=4$
D. $\log _{4} 625=5$
13. If $\log _{81} x=-\displaystyle\frac{3}{4}$ then $x=$
A. $27$
B. $\displaystyle\frac{1}{3}$
C. $\displaystyle\frac{1}{27}$
D. $\displaystyle\frac{1}{9}$
14. If antilog $3.8716=7440$ and $\log x=0.8716$ then $x=$
A. $74.40$
B. $7.440$
C. $744.0$
D. $7440$
15. If $\log 5=0.6990$ and $\log 3=0.4771$, then $\log 45=$
A. $1.6532$
B. $1.1761$
C. $1.8751$
D. $1.2219$
16. 3 log2 $-2$ log5 in the simplified form is
A. $\log \displaystyle\frac{6}{10}$
B. $\log \displaystyle\frac{9}{12}$
C. $\log \displaystyle\frac{8}{25}$
D. $\log \displaystyle\frac{25}{8}$
17. If $\log _{x} 81=4$ then $x=$
A. $3$
B. $2$
C. $-1$
D. $0$
18. If $\log _{8} x=\displaystyle\frac{2}{3}$ then $x=$
A. $2$
B. $4$
C. $3$
D. $-1$
19. $3 \log 2+\log 3=\log x$ then $x=$
A. $12$
B. $18$
C. $24$
D. $30$
20. If $\log _{4} 64=x$, then $x=$
A. $2$
B. $-1$
C. $0$
D. $3$
21. If $\log _{81} 9=x$, then $x=$
A. $3$
B. $2$
C. $1$
D. $\displaystyle\frac{1}{2}$
22. If $\log_{x} 49=2 ; x=$
A. $6$
B. $3$
C. $7$
D. $0$
23. If $\log 35+\log 36=\log (3 x)$ then $x=$
A. $400$
B. $420$
C. $520$
D. $600$
24. $\log 3+\log 6-\log 2=\log x$ then $x=$
A. $10$
B. $9$
C. $8$
D. $7$
25. $\log_{x} 36=2$ then $x=$
A. $5$
B. $8$
C. $2$
D. $6$
26. If $\log_{8} 16=x$, then $x=$
A. $\displaystyle\frac{4}{3}$
B. $\displaystyle\frac{1}{2}$,
C. $2$
D. $-\displaystyle\frac{1}{2}$
27. $\log 5+\log 8-\log 6=$
A. $\log 7$
B. $\log \displaystyle\frac{13}{6}$
C. $\log \displaystyle\frac{40}{6}$
D. $\log 50$
28. The characteristic of $\log 0.00329$ is
A. $\overline{1}$
B. $\overline{3}$
C. $\overline{2}$
D. $0$
29. The characteristic of $\log 1.02$ is
A. $1$
B. $\overline{3}$
C. $0$
D. $-1$
30. If $\log _{10} 100=x$, then $x=$
A. $2$
B. $1$
C. $0$
D. $-1$
31. The characteristic of $\log 0.000753$ is
A. $\overline{1}$
B. $\overline{2}$
C. $\overline{3}$
D. $\overline{4}$
32. The exponential form of $y=\log _{a} x$ is
A. $x=y$
B. $a^{y}=x$
C. $x^{y}=a$
D. $a=x^{y}$
33. The logarithmic form of $a=y^x$ is
A. $\log _{a} x=y$
B. $\log _{y} a=x$
C. $\log _{x} y=a$
D. $\log _{a} y=x$
34. The integral part of logarithm is called
A. determinant
B. matrix
C. mantissa
D. characteristic
35. The decimal part of logarithm is called
A. determinant
B. set
C. mantissa
D. characteristic
36. $\displaystyle\frac{\log 5}{\log 3}=$
A. $\log 5-\log 3$
B. $\log_{3} 5$
C. $\log_{5} 3$
D. $\log\displaystyle\frac{ 5}{3}$
37. $\log 729=$
A. $\log 3$
B. $6 \log 3$
C. $\log 6$
D. $\log 3+\log 6$
38. If $\log 2=0.3010 . \log 3=0.4771, \log 5=0.6990$ then $\log 30=$
A. $1.4771$
B. $0.4771$
C. $-0.4771$
D. $-1.4771$
39. If $\log 2=0.3010, \log 3=0.4771$ then $\log 4.5=$
A. $0.7781$
B. $0.1761$
C.$0.6532$
D. $1.6532$
40. If $\log _{10} 7=a$, then $\log _{10}\left(\displaystyle\frac{1}{70}\right)=$
A. $-(1+a)$
B. $\displaystyle\frac{1}{1+a}$
C. $\displaystyle\frac{a}{10}$
D. $\displaystyle\frac{1}{10 a}$
41. $\displaystyle\frac{1}{\log _{a} b} \times \displaystyle\frac{1}{\log _{b} c} \times \displaystyle\frac{1}{\log _{c} a}=$
A. $-1$
B. $0$
C. $1$
D. $a b c$
42. If $\log _{10} 2=a$ and $\log _{10} 3=b$ then $\log _{5} 12=$
A. $\displaystyle\frac{a+b}{1+a}$
B. $\displaystyle\frac{2 a+b}{1+a}$
C. $\displaystyle\frac{a+2 b}{1+a}$
D. $\displaystyle\frac{2 a+b}{1-a}$
43. If $\log _{a}(a b)=x$, then $\log _{b}(a b)=$
A. $\displaystyle\frac{1}{x}$
B. $\displaystyle\frac{x}{x+1}$
C. $\displaystyle\frac{x}{1-x}$
D. $\displaystyle\frac{x}{x-1}$
44. $2 \log _{10} 5+\log _{10} 8-\displaystyle\frac{1}{2} \log _{10} 4=$
A. $2$
B. $4$
C. $2\left(1-\log _{10} 1\right)$
D. $4\left(1-\log _{10} 1\right)$
45. If $\log _{5}\left(x^{2}+x\right)-\log _{5}(x+1)=2$, then $x=$
A. $5$
B. $10$
C. $25$
D. $\displaystyle\frac{1}{5}$
46. If $\log _{10} x-5 \log _{10} 3=-2$, then $x=$
A. $\displaystyle\frac{80}{100}$
B. $\displaystyle\frac{81}{100}$
C. $\displaystyle\frac{125}{100}$
D. $\displaystyle\frac{243}{100}$
47. If $\log _{3} x+\log _{9} x^{2}+\log _{27} x^{3}=9$, then $x=$
A. $3$
B. $9$
C. $27$
D. $\displaystyle\frac{1}{3}$
48. If $a=\log _{8} 225$ and $b=\log _{2} 15$, then $\displaystyle\frac{a}{b}=$
A. $\displaystyle\frac{1}{3}$
B. $\displaystyle\frac{2}{3}$
C. $\displaystyle\frac{3}{2}$
D. $3$
49. If the logarithm of a number is $-3.153$, what are characteristic and mantissa?
A. characteristic $=-4, \quad$ mantissa $=0.847$
B. characteristic $=-4, \quad$ mantissa $=0.153$
C. characteristic $=4, \quad$ mantissa $=-0.847$
D. characteristic $=-3, \quad$ mantissa $=-0.153$
50. If $\log \left(\displaystyle\frac{a}{b}\right)+\log \left(\displaystyle\frac{b}{a}\right)=\log (a+b)$, then
A. $a+b=1$
B. $a-b=1$
C. $a=b$
D. $a^{2}+b^{2}=1$

$\begin{array}{|ll|ll|ll|ll|ll|} \hline 1. &\text {D} & 2. &\text {B} & 3. &\text {D} & 4. &\text {A} & 5. &\text {D} \\ \hline 6. &\text {C} & 7. &\text {B} & 8. & \text{C} & 9. &\text {A }& 10.&\text {D} \\ \hline 11.&\text {D} & 12.&\text {C} & 13.&\text {C} & 14.&\text {B} & 15.&\text {A} \\ \hline 16.&\text {C} & 17.&\text {A} & 18.&\text {B} & 19.&\text {C} & 20.&\text {D} \\ \hline 21.&\text {D} & 22.&\text {C} & 23.&\text {B} & 24.&\text {B} & 25.&\text {D} \\ \hline 26.&\text {A} & 27.&\text {C} & 28.&\text {B} & 29.&\text {C} & 30.&\text {A} \\ \hline 31.&\text {D} & 32.&\text {B} & 33.&\text {B} & 34.&\text {D} & 35.&\text {C} \\ \hline 36.&\text {B} & 37.&\text {B} & 38.&\text {A} & 39.&\text {C} & 40.&\text {A} \\ \hline 41.&\text {C} & 42.&\text {D} & 43.&\text {D} & 44.& \text {A}& 45.&\text {C} \\ \hline 46.&\text {D} & 47.&\text {C} & 48.&\text {B} & 49.&\text {A} & 50.&\text {A} \\ \hline \end{array}$