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mathsExponentsExponents and Logarithm Arithmetics

### Arithmetics of Logarithms

Arithmetics with logarithms, without evaluating the logarithm, is explained. For example log_a b^m = m log_a b.

The list of formulas are derived using the first principles of logarithms.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.

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What is a logarithm?

• logarithm of a number is the power with which the base is raised to get the number
• logarithm is an inverse of exponent
• both the above
• both the above

The answer is "both the above"

What is the value of log_2 8?

• by first principles of logarithm log_2 8 = log_2 (2^3) = 3
• by first principles of logarithm log_2 8 = log_2 (2^3) = 3
• 2xx8 = 16

The answer is "log_2 8 = log_2 (2^3) = 3".

By first principles, the value is expressed as an exponent of base
log_2 8 = log_2 2^3

The exponent, to which base is raised to get a number, is the logarithm to the base of the number.
log_2 8 = log_2 2^3 = 3
2 is the base
8 is the number
3 is the log of the number

What is the exponent to which 2 is raised to get value 8?

• log_2 8
• log_2 8
• 2xx8

The answer is "log_2 8"

We learned the first principles of logarithm as, for any value x and base b,
log_b x = n hArr b^n = x

In this topic, the above is used to understand some known results.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.

It is noted that 2^4 = 16 and (-2)^4 = 16, so 4th root of 16 has more than one possible values.
That is generalized as p^n = q^n does not imply that p=q.

If the base 2 is the same, then to get the value 16, the exponent is unique 4. That is only 2^4 = 16 for base 2 and number 16.

If log_b x = log_b y then is it true that x = y?

Note that by first principles,
color(coral)(log_b x = n hArr b^n=x)
color(deepskyblue)(log_b y = n hArr b^n=y)

So,
color(coral)(b^n)  = color(deepskyblue)(b^n)
color(coral)(x)  = color(deepskyblue)(y)

• No, x and y need not be equal
• yes,x=y
• yes,x=y

The answer is "if base of the logarithm is same, then x=y".

Generalizing this, for real numbers x, y and b,
log_b x = log_b y hArr x = y

What is log_2 2^10?

Note that the value is given as an exponent of base.

• 10
• 10
• 1024

The answer is "10".

Generalizing this, for real numbers b and n,
log_b b^n = n

What is 2^(log_2 8)?

Note that by first principles,
color(coral)(log_2 8 = 3)
color(deepskyblue)(2^3 = 8)

So,
color(deepskyblue)(2)^(color(coral)(log_2 8)) = 2^3
The question is, what is the simplified form of the above expression?

• 2^8
• 8
• 8

The answer is "8".

Generalizing this, for real numbers x and b,
b^(log_b x) = x

What is log_2 1?

Note that 2^0 = 1

• 1
• 0
• 0

The answer is "0".

Generalizing this, for a real number b,
log_b 1 = 0

What is log_2 2?

Note that 2^1 = 2

• 1
• 1
• 0

The answer is "1".

Generalizing this, for a real number b,
log_b b = 1

It is given that log_2 32 = 5 and log_2 16 = 4.

What is log_2 (32 xx 16)?

Note that by first principles,
color(coral)(log_2 32 = 5 hArr 2^5=32)
color(deepskyblue)(log_2 16 = 4 hArr 2^4 =16)

So,
log_2 (color(coral)(32) xx color(deepskyblue)(16) )
= log_2 (color(coral)(2^5) xx color(deepskyblue)(2^4) )
substituting the known result 2^5 xx 2^4 = 2^(5+4)
= log_2 2^(color(coral)(5) + color(deepskyblue)(4) )

• 5+4
• 5+4
• 5 xx 4

The answer is "5+4".

That is log_2 (color(coral)(32) xx color(deepskyblue)(16) ) = log_2 (color(coral)(32) )+ log_2 (color(deepskyblue)(16) )

Generalizing this, for real numbers x, y and b,
log_b(xy) = log_b x + log_b y

It is given that log_2 32 = 5 and log_2 16 = 4.

What is log_2 (32 -: 16)?

Note that by first principles,
color(coral)(log_2 32 = 5 hArr 2^5=32)
color(deepskyblue)(log_2 16 = 4 hArr 2^4 =16)

So,
log_2 (color(coral)(32) -: color(deepskyblue)(16) )
log_2 (color(coral)(2^5) -: color(deepskyblue)(2^4) )
substituting the known result 2^5 -: 2^4 = 2^(5-4)
log_2 2^(color(coral)(5) - color(deepskyblue)(4) )
The question is, what is the simplified form of the above expression?

• 5-4
• 5-4
• 5 -: 4

The answer is "5-4".

That is log_2 (color(coral)(32) -: color(deepskyblue)(16) ) = log_2 (color(coral)(32) )- log_2 (color(deepskyblue)(16) )

Generalizing this, for real numbers x, y and b,
log_b(x/y) = log_b x − log_b y

What is log_2 (8^5)?

Note that by first principles,
color(coral)(log_2 8 = 3 hArr 2^3=8)

log_2 color(coral)(8)^5
= log_2 (color(coral)(2^3))^5)
= log_2 2^(color(coral)(3)xx5)
= color(coral)(3) xx 5
= 5 xx color(coral)(3)
substituting that log_2 8 = 3
=5 xx color(coral)(log_2 8)

• 5 log_2 8
• 5 log_2 8
• log_2 (5xx8)

The answer is "5 log_2 8".

Generalizing this, for real numbers x,n and b,
log_b x^n = n log_b x

It is noted that
b^n = x hArr log_b x = n

color(coral)(2^3 = 8) hArr root(3)(8) = color(deepskyblue)(8^(1/3)= 2)
There are two exponents in the above color(coral)(2^3) and color(deepskyblue)(8^(1/3))

applying logarithm for the exponent color(coral)(2^3 = 8) with b=2 and x=8, we get
color(coral)(log_2 8 = 3)

applying logarithm for the exponent color(deepskyblue)(8^(1/3)= 2) with b=8 and x =2
color(deepskyblue)(log_8 2 = 1/3)

Comparing the two logarithms, which of the following is correct?

• log_2 8 = log_8 2
• log_2 8 = 1 // log_8 2
• log_2 8 = 1 // log_8 2

The answer is "log_2 8 = 1 // log_8 2".

Generalizing this, for real numbers a and b,
log_b a = 1 -: (log_a b)

It is noted that

color(coral)(2^12) = color(deepskyblue)(4^6)

Converting the left hand side color(coral)(2^12) to log form
color(coral)(log_2 (2^12) = 12)

The same in right hand side
color(coral)(log_2 (4^6)) = color(deepskyblue)(6) xx color(coral)(log_2(4))

Converting the right hand side color(deepskyblue)(4^6) to log form
color(deepskyblue)(log_4 (4^6) = 6)

substituting the value 6
color(coral)(log_2 (4^6)) = color(deepskyblue)(log_4 (4^6)) xx color(coral)(log_2(4))

What does the above prove?

• log_2(2^12) = log_4(2^12) xx log_2 4
• log_2(2^12) = log_4(2^12) -: log_4 2
• both the above
• both the above

The answer is "both the above".

Generalizing this, for real numbers x, a and b,
log_b x = log_a x xx log_b a
log_b x = log_a x -: log_a b

Known results in Logarithms :

log_b x = n hArr b^n = x

log_b x = log_b y hArr x = y

log_b b^n = n

b^(log_b x) = x

log_b 1 = 0

log_b b = 1

log_b(xy) = log_b x + log_b y

log_b(x/y) = log_b x − log_b y

log_b x^n = n log_b x

log_b x = (log_a x) -: (log_a b)

log_b x = log_b a xx log_a x

Solved Exercise Problem:

What is log_3 (3^(-2))?

• 1/9
• -2
• -2

The answer is "-2"

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