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Investigating types of logarithms

Extracts from this document...

Introduction

Table 1: Adding logarithms

Expression

Value

image00.png

 +image00.png

4

image00.png

4

image00.png

 + image00.png

5

image00.png

5.0118

image00.png

 + image00.png

0

image00.png

0

image00.png

 + image00.png

5

image00.png

5

image00.png

 + image00.png

1

image00.png

Undefined

image00.png

 + image00.png

4

Examples




The general statement would therefore be
image01.png

 =image00.png

, this will be validated in the following tables

Validation of general rule by examining the base*

Base

image01.png

Value of image01.png

image00.png

Values

Same negative integer

image01.png

Non-real calculation

Same fractional integer

image02.png

-6.6438

image03.png

-6.6438

Same irrational interger

image01.png

10

image00.png

10

Different positive integer

image01.png

4.2146

image00.png

4.8613

image00.png

3.7625

Zero

image01.png

Undefined

image01.png

Undefined

*? is used to represent any value

Validation of general rule through the examination of the number while keeping the

...read more.

Middle

4

1 Negative
1 positive

image01.png

 Non-real calculation

Both negative integers

image01.png

Non-real calculation

Different fractional integer

image04.png

-0.7075

image00.png

-0.7075

Same fractional integer

image04.png

-0.4150

image00.png

-0.4150

Additive Inverses

i.e. 5 + (-5) = 0

image01.png

Non-real calculation

Different irrational number

image01.png

0.7153

image00.png

0.7153

Same irrational number

image01.png

0.4306

image00.png

0.4306

Number is 0

image01.png

Undefined

Number is

image01.png

0

image00.png

0

It is important to take note of the scope and limitations of the general statement, to find out in what condition the general statement does and does not work.

image01.png

 =image00.png

The base cannot be negative, must be above zero, but cannot equal to 1.

The number cannot be negative, must be above zero, can equal to 1.

Therefore, in mathematical terms, the domains are
Base –
a: 0<a≤∞

 1
Number – M: 0<M≤∞

        N: 0<N≤∞

Only works if the base is the same.

Table 2: Subtracting logarithms

image01.png

-1.0000

image00.png

-1.0000

image01.png

2.0000

image00.png

2.0000

image01.png

0.0000

image00.png

-1.0000

...read more.

Conclusion

(-,). This clearly supports the scope and limitations of the numbers from my previous general statements which are Number – M: 0<M≤∞

y log x = log xy

This can be proved algebraically through substitution.
Let  log
ax = y 
Lets times C to both sides
C

logax = y

 C


log 
a xn = C

y
Therefore,
C
 loga x = log a xn


...read more.

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