1. Ch. 3.2: Logarithms and their Graphs What are they?
Inverse of the exponential functions
Definition: For x > 0, a > 0, and a ≠ 1, y = logax if and only if ay = x
***y = logax is equivalent to ay = x, thus a logarithm is an exponent ***
Ex: log28 = 3 is the same as 23 = 8
Ex: log = -4 is the same as 3-4 =
Ex: Find x if log4x=2
x = ay, so
x = 42 = 16

2. Common log function: a = 10
“Log” key on calculator is log, base 10
Just enter “log” then the number when working with base 10
Calculate
1. log log log10-2
1.7324
0.3617
Not possible

3. PROPERTIES Logarithm (log) loga1=0 because a0 = 1
logaa=1 because a1 = a
logaax = x because ax = ax
logax = logay, then x = y
Natural Log (ln)
1. logex = ln x
2. y = ln x if and only if x = ey

4. Finding the domain of logax
Take the x-value set it x > 0 and solve
**REMEMBER THE DOMAIN AFFECTS WHERE THE GRAPH WILL OCCUR**
Ex: f(x) = log4(1-x) Ex: g(x) = log3x2
Ex: f(x) = ln(x – 2) Ex: g(x) = ln x3
x2 > 0
1 – x > 0
All numbers work
Except 0!!
x < 1
x – 2 > 0
x3 > 0
X > 0
x > 2

5. Domain: (0, ∞) Range: (-∞, ∞) X-intercept: (1, 0) V.A.: x = 0 (y-axis)
Graphing logs
**IMPORTANT: Log functions are the inverses of exponential funcs. **reflected about the line y = x**
Easiest method for graphing without a calculator
Write the exponential function y=ax
Find its ordered pairs, then switch the pairs.
Multiply y*coefficient (if any) then plot points.
Apply any shifts to these pts. then draw curve/asymptote.
**Shifts can occur just like exponential functions, but remember that the domain can be limited**
Parent graph
Domain: (0, ∞) Range: (-∞, ∞)
X-intercept: (1, 0) V.A.: x = 0 (y-axis)

6. Ex: Graph f(x) = log2x f(x)=log2x f(x)=2x
Change to exponential function
Create a table of values
Flip the x and y values
Graph the new set of values
f(x)=log2x
f(x)=2x x -2 -1 1 2 y
.25 .5 1 2 4 x
.25 .5 1 2 4 y -2 -1 1 2
y=2x
y=log2x
*New asymptote is x = 0