Exponentiation is repeated multiplication.
Root is the inverse operation of exponentiation.
Logarithm is a new inverse operation of exponentiation.
Expansion of Numbers and Operations
○ "square root of x" -> x ∨ 2, "x root 2"
○ "log base b of x" -> x ∀ b, "x log b"
41 Logarithms Part 1
When we say log base b of x equals y, we are saying b to the y equals x.
With logs, the base of the log raised to the power of what's on the other side of the equal sign will equal the number that the log is operating on.
Change of base
Log base b of n.
What power should I put in the exponent of b so b to that power will give me n.
"the log, base b, of x"
"To what power must b be raised, in order to yield x?"
○ 1 ∀ b = 0,
The power that the base must be raised to in order to yield 1 is the 0th.
○ b ∀ b = 1,
The power that the base must be raised to in order to yield the base itself is the 1st.
○ (b∧x) ∀ b = x,
The power that the base must be raised to in order to yield the base raised to the power of a number is the power of the number.
○ b ∧ (b∀x) = x,
A number raised to the power that the number must be raised to in order to yield x is x.
In the same way that division is the inverse of multiplication a logarithm is just the inverse of exponentiation.
Anytime you see log base b of some number n, you can think about it as asking the question "What power do I need to put in my exponent to get b to that power equal to this number n?"
By definition, when we say log base b of x equals y, that's the same thing as saying "b to the y equals x".
So if you want to find log base b of x, you're asking "what power do you have to raise b to in oder to get x?"