The document summarizes the key properties of logarithmic functions in 3 sentences: Logarithmic functions have a vertical asymptote at x=0 when b>1 and are undefined for negative values of x, they are symmetric about the y-axis and decrease without bound as x approaches 0 from the right or increases without bound as x approaches 0 from the left when 0<b<1. The graph can be shifted horizontally by changing the argument of the logarithm from x to x+c or x-c, and vertically by multiplying the entire function by a constant.

1. Graph of
Logarithmic functions

2. Graph of y = log x b >1
b
2
1
1/b
1 b b2
-1 If x = ------ , then y = ------
1/b -1
1 0
b 1
b2 2

3. Sketch the graph of y = log |x| b >1
b
y = log (-x) if x < 0 y = log x if x > 0
b b
2
1
-b2 -b -1 1 b b2
If x = ------ , then y = ------ If x = ------ , then y = ------
-1 0 1 0
-b 1 b 1
-b2 2 b2 2
Graph of y = log |x| b>1
b
is symmetric with respect to y-axis, that is, logb|x| is an even function.

4. Graph of y = log x b<1
b
If x = ------ , then y = ------
b 1
1 0
1/b -1
1
1/b2 -2
1 1/b 1/b2
b
-1
-2

5. Sketch the graph of y = log |x| b<1
b
y = log (-x) if x < 0 y = log x if x > 0
b b
If x = ------ , then y = ------ If x = ------ , then y = ------
-1 0 1 0
-1/b -1 1/b -1
-1/b2 -2 1/b2 -2
-1/b2 -1/b -1 1 1/b 1/b2
-1
-2
Graph of y = log |x| b<1
b
is symmetric with respect to y-axis, that is, logb|x| is an even function.

6. Graph of y = log (x+3)
5
1
-3 -2 2
If x = ------ , then y = ------
-2 0
2 1

7. Graph of y = log (x-3)
0.5
If x = ------ , then y = ------
4 0
3.5 1
1
3.5
3 4

8. Graph of y = -log (x+3)
0.5
1
-3 -2 -1
If x = ------ , then y = ------
-2 0
-1 1