1. 11.4 Logarithmic Functions
Objectives: Evaluate expressions involving logarithms. Solve equations and inequalities with logs. Graph logarithmic functions and inequalities.

2. Exponential Functions and Logarithmic Functions are Inverses.
2^x=y y -1
2^-1=y 2
2^2=y 6
2^6=y y
2^y=x x
2^y=½
2^y=4 4
2^y=64 64
The exponent y is called the logarithm, base 2 of x, written as: log2x = y It is read, “the log base 2 of x is = to y”

3. A log is an EXPONENT!!!! Exponential Equation Logarithmic Equation
x = a^y y = logax
x→number
a → base
y → exponent
Definition of The logarithm function y = logax, where
Logarithm: a>0 and a≠1, is the inverse of the
exponential function y=a^x.
SO, y=logax if and only if x = a^y.

4. Ex. 1)
Exponential Equation Log Equation
a.) 3²=9 a.) log39=2
b.) 10^4=10,000 b.) log1010,000=4
c.) 4^0=1 c.) log41=0
d.) 3^-3=1/27 d.) log31/27=-3
e.) 25^½=5 e.) log255=½
Ex. 2)
Evaluate:
a.) log71/49=y
b.) log⅓27=y
c.) log6x=2

5. Composites * logaa^x * a^logax=x
Properties
of Logs:
Property Definition
Product logbmn = logbm + logbn
Quotient logbm/n = logbm – logbn
Power logbm^p = p · logbm
Equality If logbm = logbn, then m=n

6. Ex. 3)
Evaluate
a.) log55^8 b.) 5^(log5(2x+2))
Ex. 4)
Evaluate
a.) log7(2x+1) = log7(3x-5)
b.) log8(x²-14) = log8(5x)
Ex. 5)
Evaluate
a.) logb15,625^1/6) = -⅓
b.) log3(4x+5) – log3(3-2x) = 2

7. Ex. 6)
Graph y = log2(x-1)
Ex. 7)
Graph y < log3(x+1)