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# Investigating types of logarithms

Extracts from this document...

Introduction

 Expression Value + 4 4 + 5 5.0118 + 0 0 + 5 5 + 1 Undefined + 4

Examples

The general statement would therefore be = , this will be validated in the following tables

Validation of general rule by examining the base*

 Base Value of  Values Same negative integer Non-real calculation Same fractional integer -6.6438 -6.6438 Same irrational interger 10 10 Different positive integer 4.2146 4.8613 3.7625 Zero Undefined Undefined

*? is used to represent any value

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

Middle

4

1 Negative
1 positive Non-real calculation

Both negative integers Non-real calculation

Different fractional integer -0.7075 -0.7075

Same fractional integer -0.4150 -0.4150

i.e. 5 + (-5) = 0 Non-real calculation

Different irrational number 0.7153 0.7153

Same irrational number 0.4306 0.4306

Number is 0 Undefined

Number is 0 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. = 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 -1 -1 2 2 0 -1

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

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