Download PPT 6 - Cobb Learning

Let’s say I give you f x = 3𝑥
I want you to find the inverse of this function
Logarithms
Logarithms
Exponential Equations:
𝑏
Logarithmic Equations:
𝑥
Base
Exponent
𝑏
𝑎
𝑥
Base
What it equals
Exponent
Reading Logarithms
▪ You read
𝑏
𝑎
𝑥 as: “the log of base b of a is x.
▪ Another way to say this is “the log is the exponents.”
▪ Just like in exponential equations, b > 0, b ≠ 1.
Example 1: Write the exponential
equation in logarithmic form
26 = 64
You Try 1:Write the exponential equation
in logarithmic form
45 = 1024
Example 2: Write the logarithmic
equation in exponential form
log 5 125 = 3
▪ A logarithm with base 10 is called a common logarithm. If no base is
written for the logarithm, the base is assumed to be 10.
▪ Ex: log 4
▪ Special properties of logarithms:
Logarithmic Form
Exponential Form
Example
Logarithm of base b:
log 𝒃 𝒃 = 𝟏
𝑏1 = 𝑏
log 𝟏𝟎 𝟏𝟎 = 𝟏
101 = 10
Logarithm of 1:
log 𝒃 𝟏 = 𝟎
𝑏0 = 1
log 𝟏𝟎 𝟏 = 𝟎
100 = 1
Evaluating Logarithms
▪ When it comes to evaluating logarithms, ask yourself this question,
“what raised to the x power gives me this value?”
▪ Then decide what x is
Example 3: Evaluate
log 1000
You Try 2: Evaluate
1
log 4
4
Example 4:Evaluate
log1000 1000
Using Inverses
▪ log 𝑏 𝑏 𝑚 = 𝑚
– m can be a numeric value or an expression
▪ 𝑏 log𝑏 𝑚 = 𝑚
– m can be a numeric value or an expression
▪ When your bases aren’t the same, manipulate the base to help you.
Example 5: Simplify the expression
5log5 𝑥
You Try 3: Simplify the expression
log 2 2𝑥
Example 6:
log 4 16𝑥
Graphing Logarithms
▪ Remember earlier in the lesson I told you logarithms were the inverse
of exponentials.
▪ When it comes to graphing logarithms, make a table of the logarithm
in exponential form and switch your x and y values.
▪ Furthermore everything vertical becomes horizontal and everything
horizontal becomes vertical (this is in respects to your asymptotes).
Example 6: Find the inverse of the
following and graph the inverse
▪ 𝑓 −1 (𝑥) = log 3 𝑥 − 1 − 1
𝑓 𝑥 = 3(𝑥+1) + 1
-1
𝑓 𝑥 = 3(𝑥+1) + 1
4
3
2
0
4
1
10
2
28
x
-2
x
𝑓 −1 (𝑥) = log 3 𝑥 − 1 − 1
4
3
2
−2
4
0
10
1
28
2
−1
Homework:
Page 277 17-30 all
(29-30 don’t describe the domain and
range of each function)