Chapter 5: Exponentials & Logarithms
Revision:
𝑎 = 𝑛 ⟺ log 𝑛 = 𝑥
Laws of Logarithms:
1. log (𝑥𝑦) = log 𝑥 + log 𝑦
2. log = log 𝑥 − log 𝑦
3. log 𝑥 = 𝑛 log 𝑥
4. log = log 𝑥 = − log 𝑥
5. log 𝑏 = the base
change
6. log 1 = 0
7. log 𝑎 = 1 85
8. Change base log 𝑏 =
Exponential function:
a a
𝑦=𝑎 0 y
9 03 on a a
16
xo.EE E
Note: Asymptote: 𝑦 = 0
Passing through (0,1) x 0 y a 1
, Example 1:
a) On the same axes, sketch the graphs of 𝑦 = 3 , 𝑦 = 2 and 𝑦 = 1.5
y 3
b) On another graph, sketch the graphs of 𝑦 = and 𝑦 = 2
9 2
5 137 my
t t.tl
2 2
9 E
Example 2:
Sketch the graph of 𝑦 = and give the coordinates of the point where
the graph crosses the y-axis. 3 1 3
y 1 2
ay
X
g.nl 8.2
Revision:
𝑎 = 𝑛 ⟺ log 𝑛 = 𝑥
Laws of Logarithms:
1. log (𝑥𝑦) = log 𝑥 + log 𝑦
2. log = log 𝑥 − log 𝑦
3. log 𝑥 = 𝑛 log 𝑥
4. log = log 𝑥 = − log 𝑥
5. log 𝑏 = the base
change
6. log 1 = 0
7. log 𝑎 = 1 85
8. Change base log 𝑏 =
Exponential function:
a a
𝑦=𝑎 0 y
9 03 on a a
16
xo.EE E
Note: Asymptote: 𝑦 = 0
Passing through (0,1) x 0 y a 1
, Example 1:
a) On the same axes, sketch the graphs of 𝑦 = 3 , 𝑦 = 2 and 𝑦 = 1.5
y 3
b) On another graph, sketch the graphs of 𝑦 = and 𝑦 = 2
9 2
5 137 my
t t.tl
2 2
9 E
Example 2:
Sketch the graph of 𝑦 = and give the coordinates of the point where
the graph crosses the y-axis. 3 1 3
y 1 2
ay
X
g.nl 8.2
