# Log Rules OR Logarithm Rules

Updated: 20 Aug 2023

2098

## Log Rules

Rules of logarithm also refer to “Log Rules”. This comprehensive guide will explore the basic log rules used to solve complex logarithm equations and problems. Some of the log rules are mentioned below:

1. Product Log Rule
\log _a m n=\log _a m+\log _a n
2. Quotient Log Rule
\log _a \frac{m}{n}=\log _a m-\log _a n
3. Power Log Rule
\log _a m^n=n \log _a m
4. Change of Base Law Rule
\log _a m \log _m n=\log _a n
\log _m n=\frac{\log _a n}{\log _a m}
5. Identity Log Rule
\log _a a=1
\log _{10} 10=1
6. Zero Log Rule
\log_a ⁡ 1=0

## Proof of Log Rules

Following are the detailed proof of the log rules.

### Product Log Rule

The logarithm of two products equals the sum of the individual logarithms.

#### Proof of Product Log Rule

Let \log _a m=x \ and \ \log _a n=y
Write them in Exponential form:
a^x=m \ and \ a^y=n
Now multiply these:
a^x \times a^y=mn
Or
mn=a^x \times a^y
mn=a^{x+y}
Taking \log _a on B.S
\log _a m n=\log _a a^{x+y}
\log _a m n=(x+y) \log _a a
\log _a m n=(x+y)(1) \qquad \log _a a=1
\log _a m n=x+y
\log _a m n=\log _a m+\log _a n

### Quotient Log Rule

The logarithm of a quotient is equal to the difference of the individual logarithms.

#### Proof of Quotient Log Rule

Let \log _a m=x \ and \ \log _a n=y
Write them in Exponential form:
a^x=m \ and \ a^y=n
Now Divide these:
\frac{a^x}{a^y}=\frac{m}{n}
Or
\frac{m}{n}=\frac{a^x}{a^y}
\frac{m}{n}=a^{x-y}
Taking \log _a on B.S
\log _a \frac{m}{n}=\log _a a^{x-y}
\log _a \frac{m}{n}=(x-y) \log _a a
\log _a \frac{m}{n}=(x-y)(1) \qquad \log _a a=1
\log _a \frac{m}{n}=x-y
Hence \ \log _a \frac{m}{n}=\log _a m-\log _a n

### Power Log Rule

The logarithm of a number raised to a power equals the exponent times the logarithm of the base:

#### Proof of Power Log Rule

Let \log _a m=x
In Exponential form:
a^x=m
Or
m=a^x
Taking power ‘ n ‘ on B.S
m^n=\left(a^x\right)^n
m^n=a^{n x}
Taking \log _a on B.S
\log _a m^n=\log _a a^{n x}
\log _a m^n=n x \log _a a
\log _a m^n=n x(1) \qquad \log _a a=1
\log _a m^n=n x
\log _a m^n=n \log _a m

### Identity Log Rule

• \log _a a=1
• \log _{10} 10=1

• \log_a ⁡ 1=0

## Log Rules MCQs

1. \log _a m n=
O \log _a m+\log _a n
O \log _a m-\log _a n
O n \log _a m
O All of them

\log _a m+\log _a n
Explanation:
Let \log _a m=x \ and \ \log _a n=y
Write them in Exponential form:
a^x=m \ and \ a^y=n
Now multiply these:
a^x \times a^y=mn
Or
mn=a^x \times a^y
mn=a^{x+y}
Taking \log _a on B.S
\log _a m n=\log _a a^{x+y}
\log _a m n=(x+y) \log _a a
\log _a m n=(x+y)(1) \qquad \log _a a=1
\log _a m n=x+y
\log _a m n=\log _a m+\log _a n

2. \log _a \frac{m}{n}=
O \log _a m+\log _a n
O \log _a m-\log _a n
O n \log _a m
O All of them

\log _a m-\log _a n
Explanation:
Let \log _a m=x \ and \ \log _a n=y
Write them in Exponential form:
a^x=m and a^y=n
Now Divide these:
\frac{a^x}{a^y}=\frac{m}{n}
Or
\frac{m}{n}=\frac{a^x}{a^y}
\frac{m}{n}=a^{x-y}
Taking \log _a on B.S
\log _a \frac{m}{n}=\log _a a^{x-y}
\log _a \frac{m}{n}=(x-y) \log _a a
\log _a \frac{m}{n}=(x-y)(1) \qquad \log _a a=1
\log _a \frac{m}{n}=x-y
Hence \ \log _a \frac{m}{n}=\log _a m-\log _a n

3. \log _a m^n=
O \log _a m+\log _a n
O \log _a m-\log _a n
O n \log _a m
O All of them

n \log _a m
Explanation:
Let \log _a m=x
In Exponential form:
a^x=m
Or
m=a^x
Taking power ‘ n ‘ on B.S
m^n=\left(a^x\right)^n
m^n=a^{n x}
Taking \log _a on B.S
\log _a m^n=\log _a a^{n x}
\log _a m^n=n x \log _a a
\log _a m^n=n x(1) \qquad \log _a a=1
\log _a m^n=n x
\log _a m^n=n \log _a m

4. \quad \log _a m+\log _a n
O \log _a \frac{m}{n}
O \log _a m n
O n \log _a m
O All of them

\log _a m n
Explanation:
\log _a m n=\log _a m+\log _a n

5. \log _a m-\log _a n
O \log _a \frac{m}{n}
O \log _a m n
O n \log _a m
O All of them

\log _a \frac{m}{n}
Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

6. n \log _a m
O \log _a \frac{m}{n}
O \log _a m n
O \log _a m^n
O All of them

Explanation:
\log _a m^n=n \log _a m

7. \log m n=
O \log m+\log n
O \log m-\log n
O n \log m
O All of them

\log m+\log n
Explanation:
\log _a m n=\log _a m+\log _a n

8. \log \frac{m}{n}=
O \log m+\log n
O \log m-\log n
O n \log m
O All of them

\log m-\log n
Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

9. \log m^n=
O \log m+\log n
O \log m-\log n
O n \log m
O All of them

n \log m
Explanation:
\log _a m^n=n \log _a m

10. \log m+\log n
O \log \frac{m}{n}
O \log m n
O \log m^n
O All of them

\log m n
Explanation:
\log _a m n=\log _a m+\log _a n

11. \log m-\log n
O \log \frac{m}{n}
O \log m n
O \log m^n
O All of them

\log \frac{m}{n}
Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

12. n \log m
O \log \frac{m}{n}
O \log m n
O \log m^n
O All of them

\log m^n
Explanation:
\log _a m^n=n \log _a m

13. \log 2 \times 3=
O \log 2+\log 3
O \log 2-\log 3
O 2 \log 3
O All of them

\log 2+\log 3
Explanation:
\log _a m n=\log _a m+\log _a n

14. \log \frac{2}{3}=
O \log 2+\log 3
O \log 2-\log 3
O 2 \log 3
O All of them

\log 2-\log 3
Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

15. \log 3^2=
O \log 2+1
O \log 2-1
O 2 \log 3
O All of them

2 \log 3
Explanation:
\log _a m^n=n \log _a m

16. \log 2+\log 3
O \log 2 \times 3
O \log 6
O \log 2
O Both a & b

\log 6
Explanation:
\log 2+\log 3 =\log 2\times 3
\log 2+\log 3 =\log 6

17. \log 2-\log 3
O \log \frac{2}{3}
O \log 2 \times 3
O \log 3^2
O All of them

\log \frac{2}{3}
Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

18. 2 \log 3=
O \log \frac{2}{3}
O \log 2 \times 3
O \log 3^2
O All of them

\log 3^2
Explanation:
\log _a m^n=n \log _a m
2 \log 3= \log 3^2
2 \log 3= \log 9

19. If \log _2 6+\log _2 7=\log _2 a \ then \ a=
O 6
O 7
O 24
O 42

42
Explanation:
\log _2 6+\log _2 7=\log _2 a
As \log _a m n=\log _a m+\log _a n
\log _2 6 \times 7=\log _2 a
\log _2 42=\log _2 a
Thus \ a=42

20. \log _a m \log _m n=
O \log _a n
O \log _a m
O Both a & b
O None of these

\log _a n
Explanation:
Let \log _a m=x and \log _m n=y
Write them in Exponential form:
a^x=m \ and \ m^y=n
Now multiply these:
As a^{x y}=\left(a^x\right)^y
But \left(a^x\right)^y=m
So a^{x y}=(m)^y=n
Then a^{x y}=n
Taking \log _a on B.S
\log _a a^{x y}=\log _a n
(x y) \log _a a=\log _a n
x y(1)=\log _a n \qquad As \ \log _a a=1
Now
\log _a m \log _m n=\log _a n

21. \log _2 3 \log _3 5=
O \log _5 2
O \log _2 5
O Both a & b
O None of these

\log _2 5
Explanation:
\log _a m \log _m n=\log _a n

22. \log _2 3 \log _3 4 \log _4 5=
O \log _5 2
O \log _2 5
O Both a & b
O None of these

\log _2 5
Explanation:
\log _a m \log _m n=\log _a n

23. \log _m n=\frac{\log _a n}{\log _a m} is called ____________ law
O Logarithm
O Change of Base
O Change of Logarithm
O None of these

Change of Base
Explanation:

24. \frac{\log _a n}{\log _a m}=
O \log _m n
O \log _t r
O \log 10
O None of these

\log _m n
Explanation:

25. \frac{\log _7 r}{\log _7 t}=
O \log _m n
O \log _t r
O \log 10
O None of these

\log _t r
Explanation:

26. \log _a a=
O 0
O 1
O 10
O None of these

1
Explanation:

27. \log _{10} 10=
O 0
O 1
O 10
O None of these

1
Explanation:

28. log⁡10= __________
O 0
O 1
O 10
O None of these

1
Explanation:

29. log_a⁡ 1= __________
O 0
O 1
O 10
O None of these

0
Explanation:

29. log⁡1= __________
O 0
O 1
O 10
O None of these

Explanation:

## Review Exercise # 3

1. \log _9 \frac{1}{81}=
O -1
O -2
O 2
O Does not exist

-2
Explanation:
\log _9 \frac{1}{81}=\log _9 \frac{1}{9^2}
\log _9 \frac{1}{81}=\log _9 9^{-2}
\log _9 \frac{1}{81}=-2 \log _9 9
\log _9 \frac{1}{81}=-2(1)
\log _9 \frac{1}{81}=-2

2. If \log _2 8=x \ then \ x=
O 64
O 3^2
O 3
O 2^8

3
Explanation:

\log _2 8=x
\log _2 2^3=x
3 \log _2 2=x
3(1)=x
3=x

3. Base of common log is:
O 10
O e
O \pi
O 5

10
Explanation:

4. \log \sqrt{10}=
O -1
O -\frac{1}{2}
O \frac{1}{2}
O 2

\frac{1}{2}
Explanation:
\log \sqrt{10} =\log (10)^{\frac{1}{2}}
\log \sqrt{10} =\frac{1}{2} \log 10
\log \sqrt{10} =\frac{1}{2}(1)
\log \sqrt{10} =\frac{1}{2}

5. For any non-zero value of x \cdot x^0=
O 2
O 1
O 0
O 10

1
Explanation:

6. Rewrite t=\log _b m as an exponent equation
O t=m^b
O b^m=t
O m=b^t
O m^t=b

m=b^t
Explanation:

7. \log _{10} 10=
O 2
O 3
O 0
O 1

1
Explanation:

8. Characteristics of \log 0.000059 is
O -5
O 5
O -4
O 4

-5
Explanation:

9. Evaluate \log _7 \frac{1}{\sqrt{7}}
O -1
O -\frac{1}{2}
O \frac{1}{2}
O 2

-\frac{1}{2}
Explanation:
\log _7 \frac{1}{\sqrt{7}} =\log _7 \frac{1}{(7)^{\frac{1}{2}}}
\log _7 \frac{1}{\sqrt{7}} =\log _7 7^{-\frac{1}{2}}
\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2} \log _7 7
\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}(1)
\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}

10. Base of natural log is
O 10
O e
O \pi
O 1

Explanation:

11. \log m+\log n=
O \log m\log n
O \log m-\log n
O \log mn
O \log \frac{m}{n}

\log mn
Explanation:

12. 0.069 can be written in scientific notation as
O 6.9 \times 10^3
O 6.9 \times 10^{-2}
O 0.69 \times 10^3
O 69 \times 10^2

6.9 \times 10^{-2}
Explanation:

13. \ln x-2 \ln y
O \ln \frac{x}{y}
O \ln x y^2
O \ln \frac{x^2}{y}
O \ln \frac{x}{y^2}