# POWER RULE OF LOGARITHMS

Before learning the power rule of logarithms, we have to be aware of the parts of a logarithm.

Consider the logarithm given below.

logba

In the logarithm above, 'a' is called argument and 'b' is called base.

## Power Rule of Logarithm

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

logamn = nlogam

In other words, if there is a value multiplied in front of a logarithm, the value can be taken as exponent to the argument.

alogxb = logxab

Apart from the power rule of logarithms, there are two other important rules of logarithm.

(i) Product Rule

(ii) Quotient Rule

### Product Rule of Logarithms

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

logamn = logam + logan

## Quotient Rule of Logarithms

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

loga(m/n) = logam - logan

## Video Lesson

### Solved Problems

Problem 1 :

Find the logarithm of 64 to the base 4.

Solution :

Write 64 as a power of 4.

64 = 4 x 4 x 4

= 43

log464 = log4(4)3

= 3log44

= 3(1)

= 3

Problem 2 :

Find the logarithm 1728 to the base 2√3.

Solution :

Write 1728 as a power of 23.

1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3

1728 = 26 x 33

1728 = 26 x [(3)2]3

1728 = 26 x (3)6

1728 = (23)6

log2√3(1728) = log2√3(2√3)6

Using the power rule of logarithms,

= 6log2√3(2√3)

= 6(1)

= 6

Problem 3 :

Find the logarithm of 0.0001 to the base 0.1.

Solution :

log0.10.0001 = log0.1(0.1)4

= 4log0.10.1

= 4(1)

= 4

Problem 4 :

Find the logarithm 1/64 to the base 4.

Solution :

log4(1/64) = log41 - log464

= 0 - log4(4)3

= -3log44

= -3(1)

= -3

Problem 5 :

Find the logarithm of 0.3333...... to the base 3.

Solution :

log3(0.3333......) = log3(1/3)

= log31 - log33

= 0 - 1

= -1

Problem 6 :

If logy(√2) = 1/4, find the value of y.

Write the equation in exponential form.

√2 = y1/4

Raise to the power 4 on both sides.

(√2)4 = (y1/4)4

(21/2)4 = y

22 = y

4 = y

Problem 7 :

Simplify :

(1/2)log1025 - 2log103 + log1018

Solution :

= (1/2)log1025 - 2log103 + log1018

Using power rule of logarithms,

= log10251/2 - log1032 + log1018

= log10(52)1/2 - log1032 + log1018

= log105 - log109 + log1018

= log105 + log1018 - log109

Using the product rule of logarithms,

= log10(5 x 18) - log109

= log1090 - log109

Using the quotient rule of logarithms,

= log10(90/9)

= log1010

= 1

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