Logarithmic Functions
MCR3U9
Recall: When solving exponential equations, we have the tools to do this only when we can
create two like bases,
1
eg. 8𝑥 =
√2
1 𝑥+1
Motivation for logarithms: How then can we solve (2)
bases?
= 3 when we can’t create like
Example 1 Graph 𝑓(𝑥) = 10𝑥
On the same axes graph 𝑦 = 𝑓 −1 (𝑥)
Recall that the inverse function is obtained by interchanging 𝑥 and 𝑦
(the graph of a function and its inverse should be the reflection over the line 𝑦 = 𝑥)
𝑓 −1 (𝑥) of an exponential function is a ___________________ function.
The definition of log:
𝑦 = 10𝑥 ⟺ 𝑥 = log10 𝑦 (𝒐𝒓 𝐥𝐨𝐠 𝒚)
In general,
𝑦 = 𝑎 𝑥 ⟺ 𝑥 = log 𝑎 𝑦
Another way of thinking about it:
𝑥 = log10 𝑦
means 𝑥 is the exponent to which 10 (the base) is raised in order to get 𝑦.
Try it yourself!
1) Write an equivalent exponential statement for log10 1000 = 3.
2) Write an equivalent logarithmic statement for 34 = 81
3) Find log10 100
Hint: “What power must 10 be raised to, to get 100?”
4) Find log 2 32
5) Find log 5 0.2
Example 2 Evaluate.
a) log 𝑎 𝑎𝑛
b) log 2
Example 3 Use a calculator to find:
a) log 1000
b) log10 52
Example 4 Find an approximate value of log 2 10
1
√2
The logarithmic function is defined as 𝑦 = log b 𝑥, or 𝑦 equals the logarithm of 𝑥 to the base 𝑏.
This function is defined only for 𝑏 > 0, 𝑏 ≠ 1.
𝑦 = log b 𝑥
𝑦 = 𝑏𝑥
↔
↔
𝑥 = 𝑏𝑦
𝑥 = log b 𝑦
Recall: Below is the graph of 𝑦 = 2𝑥 and its inverse. Write the equation of the inverse and
analyse its coordinates.
𝑦 = log 2 𝑥
Points
𝑦 = 2𝑥
Example 5
Inverse
Verify points
Graph 𝑦 = 3𝑥 and 𝑦 = log 3 𝑥 and complete the table below.
y  3x
y  log 3 x
Type of Function
Domain
Range
𝑥-intercept
𝑦-intercept
asymptote
Example 6 Write each in logarithmic form.
a) 16 = 24
b) 𝑚 = 𝑛3
1
c) 3−2 = 9
Example 7 Write each in exponential form.
a) log 4 64 = 3
b) 𝑦 = log 100
c) 𝑦 = ln 𝑥
Example 8 Evaluate each.
a) log 2 16
e) log 4 1
3
h) log 7 √7
1
b) log 3 (81)
c) log 36 6
1
f) log 5 125
i) log 9 3
d) log 9 9
g) log 2 (8)
j) log 2 (4√8)
k) log 8 4
Example 9 Evaluate to 2 decimals places when necessary. (log
a) log10 100
b) log10 1
c) log 5.4
key -> log10
d) log 23
common log)
e) ln 6
Example 10 Evaluate
a) log 4 (8√2)
1
b) log 5 (5√5)
c) log √2 32
d) log 1 16
8