Warm up
What is the inverse of the
exponential function 𝑦 =
𝑥
2 ?
Solution
• We can see the inverse is 𝑥 = 2𝑦 by switching
the variables, but how do we solve for y?
• It turns out, we need a special function called
a Logarithmic function to do this.
Why do we care?
• Logarithms are functions that used to be very
helpful, but most of their value has become
obsolete now that we have fancy calculators.
• BUT! They still are extremely important in
solving exponential equations.
• That is what we will use them for.
First we have to understand what a logarithm is
before we can use it.
Definition of a Logarithm
• Does the 𝑎 = 𝑏 𝑥 look familiar?
Let’s use this definition to solve the
warm-up.
• We have 𝑥 = 2𝑦 , and we want to solve for y.
• We are going to rewrite this function using
logs to solve for y.
• 𝑥 = 2𝑦 becomes log 2 𝑥 = 𝑦
• So 𝑓 −1 𝑥 = log 2 𝑥
What are all of these numbers?
Number
Exponent
𝑦 = log 2 𝑥
Base (if no base, then base is 10)
Example 1
• Rewrite the logarithmic equation in
exponential form using the definition. Solve
for y.
log 1000 = 𝑦
Solution
•
•
•
•
log 1000 = 𝑦 Original
1000 = 10𝑦 Using definition
103 = 10𝑦
1000=103
y=3
Equal base rule.
You try
log 5 625 = 𝑦
Look at the definition, and plug in numbers
where they belong.
Solution
•
•
•
•
log 5 625 = 𝑦
625 = 5𝑦
54 = 5 𝑦
y=4
Original
Using Definition
Get common base
Common base rule
One more, a little harder: Solve for x
4
log 9 9 = 𝑥
Solution
4
• log 9 9 = 𝑥
•
4
9 = 9x
1
4
• 9 = 9𝑥
• x=
1
4
Original
Definition of log
Write root as a radical
Common base rule.
We will also want to find exact values
of logs.
• If the log is base 10, then it is easy to do on
the calculator. If it is not base 10, we will have
to put it in terms of base 10 to use the
calculator.
Example
• Find the exact value of log 20 𝑎𝑛𝑑 log 1
• On your calculators, find the log button
(middle left). Enter log(20).
• You should get 1.3.
• What do you get for log(1)?
But, there is not a button if the base is
not 10…
We will use this rule to solve for exact values of logs on our
calculators.
Example
𝑙𝑜𝑔243
log 3 243 =
=5
𝑙𝑜𝑔3
*To get 5, simply enter the log fraction in the
calculator.
Example
• Solve
4𝑥 = 128.
Solution
• 4𝑥 = 128.
• x=log 4 128
•
𝑙𝑜𝑔128
x=
𝑙𝑜𝑔4
• X=3.5
Original
Rewrite using logs
Use change of base rule
Enter in calculator.
You try
• 6𝑥 = 12
Answer
• 6𝑥 = 12
• 𝑥 = log 6 12
• 𝑥=
𝑙𝑜𝑔12
𝑙𝑜𝑔6
• x=1.39
Homework
• 5.6 Worksheet
1:all, 2:a-e, 3:Skip, 4:skip, 5:a-g