# FUNDAMENTAL LAWS OF LOGARITHMS

In this section, you will learn about fundamental laws of logarithms.

There are three fundamental laws of logarithms.

Law 1 :

Logarithm of product of two numbers is equal to the sum of the logarithms of the numbers to the same base.

That is,

logamn  =  logam + logan

Law 2 :

Logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base.

That is,

loga(m/n)  =  logam - logan

Law 3 :

Logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number to the same base.

That is,

logamn  =  nlogam

## Fundamental Laws of Logarithms - Practice Problems

Problem 1 :

Find the logarithm of 64 to the base 2√2.

Solution :

Write 64 as in terms of 2√2.

64  =  26

64  =  24+2

64  =  2 22

64  =  2⋅ [(√2)2]2

64  =  2⋅ (√2)4

64  =  (2√2)4

Then,

log2√264  =  log2√2(2√2)4

log2√264  =  4log2√2(2√2)

log2√264  =  4(1)

log2√264  =  4

Problem 2 :

Find the value of log√264.

Solution :

log√264  =  log√2(2)6

log√264  =  6log√2(2)

log√264  =  6log√2(√2)2

log√264  =  6 ⋅ 2log√2(√2)

log√264  =  12 ⋅ 2(1)

log√264  =  12

Problem 3 :

Find the value of log(0.0001) to the base 0.1.

Solution :

log0.1(0.0001)  =  log0.1(0.1)4

log0.1(0.0001)  =  4log0.10.1

log0.1(0.0001)  =  4(1)

log0.1(0.0001)  =  4

Problem 4 :

Find the value of log (1/81) to the base 9.

Solution :

log9(1/81)  =  log91 - log981

log9(1/81)  =  0 - log9(9)2

log9(1/81)  =  -2log99

log9(1/81)  =  -2(1)

log9(1/81)  =  -2

Problem 5 :

Find the value of log(0.0625) to the base 2.

Solution :

log2(0.0625)  =  log2(0.5)4

log2(0.0625)  =  4log2(0.5)

log2(0.0625)  =  4log2(1/2)

log2(0.0625)  =  4(log21 - log22)

log2(0.0625)  =  4(0 - 1)

log2(0.0625)  =  4(-1)

log2(0.0625)  =  -4

Problem 6 :

Find the value of log(0.3) to the base 9.

Solution :

log9(0.3)  =  log9(1/3)

log9(0.3)  =  log91 - log93

log9(0.3)  =  0 - log93

log9(0.3)  =   - log93

log9(0.3)  =  - 1 / log39

log9(0.3)  =  - 1 / log332

log9(0.3)  =  - 1 / 2log33

log9(0.3)  =  - 1 / 2(1)

log9(0.3)  =  -1/2

Problem 7 :

Given log2 = 0.3010 and log3 = 0.4771, find the value of log6.

Solution :

log6  =  log(2 ⋅ 3)

log6  =  log2 + log3

Substitute the values of log2 and log3.

log6  =  0.3010 + 0.4771

log6  =  0.7781

Problem 8 :

If 2logx  =  4log3,  then find the value of 'x'.

Solution :

2logx  =  4log3

Divide each side by 2.

logx  =  (4log3) / 2

logx  =  2log3

logx  =  log32

logx  =  log9

x  =  9 After having gone through the stuff given above, we hope that the students would have understood the three fundamental laws of logarithms.

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