Properties of power and apower
After years of use, it’s difficult to mentally just redefine exponentiation, so let me go over all of the properties of power and apower with you:
Definition
- a ^ b = ba
- a v b = ln(a) / ln(b)
Order of operations
1 ^ 2 ^ 3 v 4 ^ 5 is read as (((1 ^ 2) ^ 3) v 4) ^ 5
You just go left to right. Treat ^ and v the same.
1 + 2 * 3 ^ 5 * 8 v 9 / 4 - 10 is read as 1 + (2 * (3 ^ 5) * (8 v 9) / 4) - 10
Order of operations:
- power and apower
- multiplication and division
- addition and subtraction
Power Identities
- 3 ^ a = a * a * a
- 2 ^ a = a * a
- 1 ^ a = a
- 0 ^ a = 1
- (-1) ^ a = 1 / a
- (-2) ^ a = 1 / (a * a) = 1 / 2 ^ a
- 2 ^ (a + b) = (a + b) * (a + b)
- 3 ^ (a + b) = (a + b) * (a + b) * (a + b)
- 4 ^ (a + b) = (a + b) * (a + b) * (a + b) * (a + b)
- a ^ (b * c) = a ^ b * a ^ c
- a ^ (b / c) = a ^ b / a ^ c
- (a + b) ^ 2 = a ^ 2 * b ^ 2
- (a - b) ^ 2 = a ^ 2 / b ^ 2
- a ^ (b ^ c) = (a * b) ^ c
APower Identities
- 10000 v 10 = 4
- 64 v 2 = 6
- (a * b) v c = a v c + b v c
- (a / b) v c = a v c - b v c
- a * b v c = a ^ b v c
- (1 / a) * b v c = b v (a ^ c)
- 1 / a v b = b v a
- a v c / b v c = a v b
- a v c * c v b = a v b
- a v (b ^ a) = 1 / b
Calculus identities
- (pi * i) ^ e = -1
- (2 * pi * i) ^ e = 1
- (-1) v e = pi * i
- Derivative (x ^ c) / dx = c v e * x ^ c
- Derivative (c ^ x) / dx = c * (c - 1) ^ x
- Derivative (x v c) / dx = e v c / x
- Derivative (c v x) / dx = -c v x / x ^ x v e
- Integral (x ^ c) dx = e v c * x ^ c + constant
- Integral (c ^ x) dx = (c + 1) ^ x / (c + 1) + constant
- Integral (x v c) dx = x * (x v c - e v c) + constant
- Integral (c v x) dx = c v e * li(x) + constant
Algebra with power and apower
If x ^ y = z then
- x = z v y
- y = (1 / x) ^ z
If x v y = z then
- x = z ^ y
- y = (1 / z) ^ x