chapter 4 theories

f(a) = f(b) only when a = b.

If the ordered pairs of a function g are the ordered pairs of a function f w/ the order of the coordinates reversed, then g is the inverse function of f.

The exponential function w/ base b is defined by f(x)=b^x where b > 0, b != 1, and x is a real number

• If x>0 and b is a positive constant except for 1 (b!=1), then y=log_bx iff b^y=x.
• ***********
• 1. log_bb = 1
• 2. log_b1 = 0
• 3. log_bb^x = x
• 4. b^log_bx = x

• log_bM*N = log_bM + log_bN
• ************
• let log_bM = x and log_bN = y,
• then b^x = M and b^y = N.
• b^x * b^y = M*N (multiplication prop.)
• b^x+y = MN (product law of exponents)
• log_bMN = x + y (substit. from the beginning)
• log_bMN = log_bM + log_bN (substit.)

log_b(M/N) = log_bM - log_bN

• log_bM^P = P * log_bM
• ************
• let log_bM = x
• then b^x = M
• (b^x)^P= M^P
• b^x*P = M^P
• log_bMP = xP
• log_bMP = P(log_bM) (substit.)

• If x, a, and b are positive real numbers w/ a != 1 and b != 1, then
• log_bx = log_ax
•              log_ab
• ************
• log_bx = y --> b^y = x
• log_ab^y = log_ax
• ylog_ab = log_ax
• y = log_ax 
•       log_ab
• log_bx = log_ax 
•              log_ab