Logarithm

Published: 14 Aug 2023

Suppose we are asked to find the value of $ x $, which makes $ 2^x=8 $. We can answer that $ x=3 $ like $ 2^3=8 $. However, if we a question $ 2^x=10 $ and find the value of $ x $, then it is not easy to find the value of $ x $. Thus, a logarithm is a tool which can solve such type of problem.

Definition of Logarithm

If $ a^x=y $, then the $ x $ is called the logarithm of $ y $ to the base $ a $ and written as $ log_a⁡y=x $, where $ a $ is a positive real number and $ a \neq 1 $.
We called $ log_a⁡y=x $ like the $ log $ of $ y $ to the base $ a $ equal to $ x $.

  • $ log_a⁡y=x \ and \ a^x=y $ are two different ways of expressing the same relation.
  1. $ log_a⁡y=x $is the logarithm form.
    The exponential form of $ log_a⁡y=x $ is $ a^x=y $.
  2. $ a^x=y $ is the exponential form.
    The logarithm form of $ a^x=y $ is $ log_a⁡y=x $.

Logarithm MCQs

1. The exponential form of $ \log _a y=x $ is
O $ a^y=x $
O $ y=x $
O $ a^x=y $
O $ a^x=y $
Show Answer

$ a^x=y $
Explanation:
We called $ \log _a y=x \ like \ log \ of \ y \ to \ the \ base \ a \ equal \ to \ x.$


2. The logarithm form of $ a^x=y $ is
O $ \log _a y=x $
O $ \log _a x=y $
O $ \log x $
O $ \log y $
Show Answer

$ \log _a y=x $
Explanation:
If $ a^x=y $ then the index $ x $ is called the logarithm of $ y $ to the base $ a $ and wirtthe as:
$ \log _a y=x $


3. The logarithm form of $ 2^{-6}=\frac{1}{64} $ is
O $ \log _{-6} \frac{1}{64}=2 $
O $ \log _{64} 2=\frac{1}{64} $
O $ \log _2 \frac{1}{64}=-6 $
O $ \log _a y=x $
Show Answer

$ \log _2 \frac{1}{64}=-6 $
Explanation:
General form of Conversion is:
$ a^x=y \longleftrightarrow \log _a y=x $


4. The logarithm form of $ 10^{\circ}=1 $ is
O $ \log _{10} 1=1 $
O $ \log _{10} 1=0 $
O $ \log _0 1=1 $
O $ \log _a y=x $
Show Answer

$ \log _{10} 1=0 $
Explanation:
$ a^x=y \longleftrightarrow \log _a y=x $


5. The logarithm form of $ x^{\frac{3}{4}}=y $ is
O $ \log _y x=\frac{3}{4} $
O $ \log _{\frac{3}{4}} y=x $
O $ \log _x y=\frac{3}{4} $
O All of them
Show Answer

$ \log _x y=\frac{3}{4} $
Explanation:
$ a^x=y \longleftrightarrow \log _a y=x $


6. The exponential form of $ \log _2 \frac{1}{128}=-7 $ is
O $ 2^{-7}=\frac{1}{128} $
O $ 2^{-7}=128 $
O $ -7^2=\frac{1}{128} $
O $ a^x=y $
Show Answer

$ 2^{-7}=\frac{1}{128} $
Explanation:
$ \log _a y=x \longleftrightarrow a^x=y $


7. The exponential form of $ \log _a a=1 $ is
O $ 1^a=1 $
O $ 1=1 $
O $ a^1=1 $
O None of them
Show Answer

$ a^1=1 $
Explanation:
$ \log _a y=x \longleftrightarrow a^x=y $


8. The exponential form of $ \log_a 1=0 $ is
O $ 1^a=0 $
O $ 1=1 $
O $ a^0=1 $
O $ a=1 $
Show Answer

$ a^0=1 $
Explanation:
$ \log _a y=x \longleftrightarrow a^x=y $


9. The exponential form of $ \log_4 \frac{1}{8}=\frac{-3}{2} $ is
O $ 4^{\frac{-3}{2}}=\frac{1}{8} $
O $ 4^{\frac{1}{3}}=\frac{-3}{2} $
O Both a & b
O None of them
Show Answer

$ 4^{\frac{-3}{2}}=\frac{1}{8} $
Explanation:
$ \log _a y=x \longleftrightarrow a^x=y $


10. The exponential form of $ \log _{\sqrt{5}} 125=x $ is
O $ (\sqrt{5})^x=125 $
O $ \left(5^{\frac{1}{2}}\right)^x=125 $
O $ 5^{\frac{x}{2}}=125 $
O All of them
Show Answer

$ (\sqrt{5})^x=125 $
Explanation:
$ \log _a y=x \longleftrightarrow a^x=y $


11. The exponential form of $ \log _3(5 x+1)=2 $ is
O $ 3^x=5 $
O $ 3^2=5 x+1 $
O $ 3^2=5 x $
O None of them
Show Answer

$ 3^2=5 x+1 $
Explanation:
$ \log _a y=x \longleftrightarrow a^x=y $


12. In $ \log _{\sqrt{5}} 125=x $, the Value of $ x $ is
O 5
O 125
O 6
O None of them
Show Answer

6
Explanation:
See Ex # 3.2
Q No. 3
Part (i)


13. In $ log x=-3 $, the Value of $ x $ is
O $ \frac{1}{64} $
O 64
O 4
O $ -3 $
Show Answer

$ \frac{1}{64} $
Explanation:
See Ex # 3.2
Q No. 3
Part (ii)


14. In $ \log _{81} 9=x $, the Value of $ x $ is
O $ \frac{1}{2} $
O 81
O 9
O $ -3 $
Show Answer

$ \frac{1}{2} $
Explanation:
See Ex # 3.2
Q No. 3
Part (iii)


15. In $ \log _3(5 x+1)=2 $, the Value of $ x $ is
O $ \frac{5}{8} $
O $ \frac{8}{5} $
O 5
O 2
Show Answer

$ \frac{8}{5} $
Explanation:
See Ex # 3.2
Q No. 3
Part (iv)


16. In $ \log _2 x=7 $ , the Value of $ x $ is
O 2
O 7
O 0
O 128
Show Answer

128
Explanation:
See Ex # 3.2
Q No. 3
Part (v)


17. In $ \log _x 0.25=2 $, the Value of $ x $ is
O $ \frac{5}{10} $
O $ \frac{1}{2} $
O $ 0.5 $
O All of them
Show Answer

$ \frac{1}{2} $
Explanation:
See Ex # 3.2
Q No. 3
Part (vi)


18. In $ \log _x(0.001)=-3 $, the Value of $ x $ is
O 1
O 10
O 0
O All of them
Show Answer

10
Explanation:
See Ex # 3.2
Q No. 3
Part (vii)


19. In $ \log _x \frac{1}{64}=-2 $, the Value of $ x $ is
O 64
O 2
O 8
O All of them
Show Answer

8
Explanation:
See Ex # 3.2
Q No. 3
Part (viii)


20. In $ \log _{\sqrt{3}} x=16 $, the Value of $ x $ is
O 6561
O 4
O 3
O None of these
Show Answer

6561
Explanation:
See Ex # 3.2
Q No. 3
Part (ix)

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jawad khalil
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