1. EXPONENTIAL & LOGARITHMIC GRAPHS
Exponential Functions
A function in the form f(x) = ax (a>0, a1) is called an exponential function to base a .
Consider f(x) = 2x x
f(x)
1/8 ¼ ½

2. y = 2x
The graph is like
(1,2)
(0,1)
Major Points
(i) y = 2x passes through the points (0,1) & (1,2) .
(ii) As x  y however as x  - y 0 .
(iii) The graph shows a GROWTH function.

3. Inverse of y = 2x
To obtain the graph of the inverse function f –1(x) we can swap the x & f(x) values round. ie
x 1/8 ¼ ½
f–1(x)
To obtain f –1(x) from x we must ask the question
“What power of 2 gives us…?”
This is not practical to write in a formula so we say
“the logarithm to base 2 of x”
f –1(x) = log2x
or “log base 2 of x”

4. y = log2x
The graph is like
(2,1)
(1,0)
NB: x>0
Major Points
(i) y = log2x passes through the points (1,0) & (2,1) .
(ii) As x  y but at a very slow rate and as x  0 y  - .

5. DON’T COPY - see handout !
The graph of y = ax always passes through (0,1) & (1,a)
It looks like .. Y
y = ax
(1,a)
(0,1) X

6. DON’T COPY - see handout !
The graph of y = logax always passes through (1,0) & (a,1)
It looks like .. Y
(a,1)
y = logax
(1,0) X