# Logarithm

Updated: 14 Aug 2023

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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)