Properties of exponential functions
(b^x)b^y = b^x+y
(b^x)^y = b^(xy)
b^x / b^y = b^x-y
(bc)^x = (b^x)c^x
Logs
Logarithms tell us how many times we multiply a variable by itself in order to get a different number.
2^x = 8 is called an exponential equation and can be written as
log2(8) = x // x = 3 here
General log rule
Given the exponential equation of a^x = y
the associated logarithmic equation is loga(y) = x
and vice versa
Product rule
loga(xy) = loga(x) + loga(y)
Quotient rule
logb(x/y) = logb(x) - logb(y)
Power rule
logc(x^n) = n(logc(x))
Common logarithms
log(y) = x is a common logarithm and has a base of 10.
Natural logarithms
ln(x) is a natural logarithm and has a base of e
e is Euler's number, and is equal to about 2.72
e is calculated as the sum of the infinite series, and arises in the study of compound interest.
e = 0
for (n=0; n < Inf; n++) e += 1/n!
Restricted values
In an expression loga(y):
The base a must be positive and not equal to 1
log1(1) = 0, log1(1) = 1, log1(1) = 2 // leads to impossible statements
The argument y must be positive
Properties of logarithmic functions
b != 1
logb(xy) = logb(x) + logb(y)
logb(x/y) = logb(x) - logb(y)
logb(x^y) ylogb(x)
logc(x) = logb(x) / logb(c) where c != 1
Evaluating logs
Try and equalize the bases and then apply power rules
If base > argument
log27(3) = 27^x = 3
27^x = 3
(3^3)^x = 3^1
3^3x = 3^1
3x = 1
x = 1/3
If argument is a fraction
log8(1/512)
8^x = 1/512
8^x = 1/8^3
8^x = 8^-3
x = -3
Change of base formula
loga(b) = logc(b)/logc(a)
You can pick any value you want for c, typically you would pick 10 or e
Graphing exponential functions
Like y = 2 * (1/3)^(-x+2) - 4
- Plug in a large value (100) and large negative value (-100) to determine the "end behavior"
- One should result in an infinite-esque value, the other should be the horizontal asymptote
- Plug in some easy to calculate values to plot
- Connect the points
Convert graphs of logs and exponential functions by switching x,y pairs (they are inverses of each other)