1. Def: is read as "log a to the base b", or in short as "log a base b" where a is the argument and b is the base 2. is defined iff
iff means 'if and only if' in standard mathematics
Note that cannot be written as , as can still be between 1 and 0
Why can't base be 1? In essence, Log finds the power given the value and base. If base is 1, value cannot be anything other than 1, as is still 1.
Proof: Anything raised to 1 is the same thing.
Proof: If any base raised to some power is 1, then the power must be 0
Let's multiply the equations
We can do this because we're still multiplying both sides by the same value (LHS by and RHS by ), as is equal to , i.e. the same as
Divide the equations
Let the answer be x
Can be written as
...using property 1. Since bases are the same powers can be equated
...using property 7.
Using property 1...
Let . Using property 7...
Take on both sides
Using property 8...
Divide both sides by
Using property 9,
Since in this case
Using property 7,
Unwrap using property 9...
Split the fraction
Re-wrap using property 9...
This property compiles properties 9 and 10.
Proof: Advanced, I may add it later
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