LOGARITHM

WHEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 2³.

2³ = 8.

Inversely, if we are given the base 2 and its power 8 --

2^? = 8

-- then what is the exponent that will produce 8?

That exponent is called a logarithm.  We call the exponent 3 the logarithm of 8 with base 2.  We write

3 = log₂8.

We write the base 2 as a subscript.
3 is the exponent to which 2 must be raised to produce 8.
A logarithm is an exponent.

Since

10⁴ = 10,000

then

log₁₀10,000 = 4.

"The logarithm of 10,000 with base 10 is 4."

4 is the exponent to which 10 must be raised to produce 10,000.

"10⁴ = 10,000" is called the exponential form.

"log₁₀10,000 = 4" is called the logarithmic form.

Here is the definition:

log_bx = n   means   bⁿ = x.

That  base  with that  exponent  produces x.

Example 1.   Write in exponential form:   log₂32 = 5.

 Answer.   2⁵ = 32.

Example 2.   Write in logarithmic form:  4−2  =

1 16

.

Answer.   log4

1 16

= −2.

Problem 1.   Which numbers have negative logarithms?

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Do the problem yourself first!

Proper fractions.

Example 3.   Evaluate  log₈1.

 Answer.   8 to what exponent produces 1?  8⁰ = 1.

log₈1 = 0.

We can observe that, in any base, the logarithm of 1 is 0.

log_b1 = 0

Example 4.    Evaluate  log₅5.

 Answer.   5 with what exponent will produce 5?   5¹ = 5.  Therefore,

log₅5 = 1.

In any base, the logarithm of the base itself is 1.

log_bb = 1

Example 5 .   log₂2^m = ?

 Answer.   2 raised to what exponent will produce 2^m ?   m, obviously.

log₂2^m = m.

The following is an important formal rule, valid for any base b:

log_bb^x = x

This rule embodies the very meaning of a logarithm.  x -- on the right -- is the exponent to which the base b must be raised  to produce b^x.

The rule also shows that the exponential function b^x is the inverse of the function log_bx.  We will see this in the following Topic.

Example 6 .   Evaluate  log3

1 9

.

Answer.

1 9

is equal to 3 with what exponent?

1 9

= 3−2.

log3

1 9

=

log33−2  =  −2.

Compare the previous rule.

Example 7.   log₂ .25 = ?

 Answer.   .25 = ¼ = 2⁻².  Therefore,

log₂ .25 = log₂2⁻² = −2.

Laws of Indices

When working with indices, there are three rules that should be used:

When m and n is positive integers,

1.	am × an = am + n
2.	am ÷ an = am – n   or	am an	= am – n   (m ≥ n)
3.	(am)n = am × n

This is the definition of aⁿ logical consequences, these three are the results, but formal proof are really needed.With particular examples as below, you can 'verify' them, but this are not the proof.

27× 23	=	(2 × 2 × 2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)
	=	2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
	=	210	(here m = 7, n = 3 and m + n = 10)

or,

27 ÷ 23	=	2 × 2 × 2 × 2 × 2 × 2 × 2 2 × 2 × 2
	=	2 × 2 × 2 × 2
	=	24	(again m = 7, n = 3 and m – n = 4)

Also,

(27)3	=	27 × 27 × 27
	=	221	(using rule 1)   (again m = 7, n = 3 and m × n = 21)

Below, the first rule proof is given below:

Proof

am × an	=	a × a × ... × a m of these	×	a × a × ... × a n of these
	=	a × a × ... × a × a × a × ... × a (m+n) of these
	=	am+n

In a similar way, for all positive integers m and n, the second and third rules can be shown.
Using rule 2 we can see an important result:

	xn xn	= xn – n = x0
but	xn xn	= 1,	so

x⁰ = 1

This is true for any non-zero value of x, so, for example, 3⁰ = 1, 27⁰ = 1 and 1001⁰ = 1.