1. EXPONENTIAL
AND
LOGARITHMIC
FUNCTIONS

2. e a x + b = p ln ( a x + b ) = q ln A = p e p = A
We shall look at how to solve equations of the forms:
e a x + b = p
ln ( a x + b ) = q
and,
Firstly it is essential to know the relationship between a
logarithmic and an exponential statement:
i.e. If:
ln A = p
Then:
e p = A
… and vice versa.

3. Example 1: Solve the equation 7 + e 2x + 1 = 50
ln A = p
e p = A
ln 43 – 1 2
ln 43 = 2x + 1 x =
= 1.38
( 3 sig.figs.)
Example 2: Solve the equation ln ( x – 5 ) = 7
1 + 2 ln ( x – 5 ) = 7
2 ln ( x – 5 ) = 6
ln ( x – 5 ) = 3
ln A = p
e p = A
e3 = x – 5
x = 5 + e3
= 25.1
( 3 sig.figs.)

4. Example 3: The temperature, θ° C, of a cup of coffee at time t minutes
after being made is given by θ = e – 0.04 t.
a) Find the initial temperature of the coffee.
b) Find how long it takes for the temperature of the coffee to be 50 ° C.
a) θ = e – 0.04 t
Initially, t = 0
θ = e 0 =
= 82
The temperature initially is 82° C.
b) When θ = 50,
50 = e – 0.04 t
32 = 64 e – 0.04 t
0.5 = e – 0.04 t
ln A = p
e p = A
ln 0.5 = – 0.04 t
t = 17.3
i.e. approximately 17 minutes.

5. ln A = p e p = A e q = ax + b ax + b = ln p If: Then:
Summary of key points:
If:
ln A = p
Then:
e p = A
Hence to solve an equation of the form: ln (a x + b) = q
We have:
e q = ax + b
and to solve an equation of the form: e ax + b = p
We have:
ax + b = ln p
This PowerPoint produced by R.Collins ; Updated Mar.2010