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
Therefore, through my observations, I hypothesise that my general rule will be
=
Validation of general rule through the examination of the number while keeping the bases the same positive integer
Validation of general rule by examining the base*
Base must be the same, cannot be equals to zero, negative or 1
Number can be different, cannot be negative and zero.
Base – a: 0<a≤∞
1
Number – M: 0<M≤∞
N: 0<N≤∞
3
n log x = log xn
4)
The graph of y = log10 x tells us many things. Firstly, we see that y(the power) is only negative when 0<x>1. We also see that x will never be negative as the graph’s vertical asymptote is 0. This means that x>0 for y to exist. However, there is no horizontal asymptote for this graph, therefore the range for y is (-∞,∞). 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 loga x = y
Lets times C to both sides
C
loga x = y
C
log a xn = C
y
Therefore,
C loga x = log a xn
