Download 2-2: Laws of Logs, Log Rules, Log Equations

IB Math SL Year 2
Name: ________________________
Date: _________________________
.
2-2: Laws of Logs, Log Rules, Log Equations
Key Notes
1. Laws of Logs and
Change of Base
2. Solving Equations
with Logs

What do I need to know?
Definitions for:
o Exponent
o Power
o Base
o Log

Calculation for/Process:
o Perform Log Rules
o Change bases
o Rewriting Logs to equivalent expressions

Definitions for:
o Power/exponent

Calculation for/Process:
o Solving equations with logs
Notes to Self
In this lesson we will revisit the following learning goals:
1.
2.
3.
4.
What are the laws of logs?
How do we change the base of logs?
How do we solve equations where the variable is in the exponent?
What methods do we use to solve equations that involve logs?
Station 1: Logarithms and Log rules
We can re-write Exponents as Logarithms (A logarithm is the ______________ to which a number must be raised in
order to get some other number)
If 𝑏 = 𝑎 𝑥 then log 𝑎 𝑏 = 𝑥
Examples)
𝟓𝟐 = 𝟐𝟓 is equivalent to 𝐥𝐨𝐠 𝟓 𝟐𝟓 = 𝟐
𝟐𝟑 = 𝟖 is equivalent to 𝐥𝐨𝐠 𝟐 𝟖 = 𝟑
𝟑𝒙 = 𝟗 is equivalent to 𝐥𝐨𝐠 𝟑 𝟗 = 𝒙
IB Math SL Year 2
Let’s try it!
Example
My thinking
“The log is the number I must raise the 5 to in
order to get 625.”
Evaluate log 5 625
1
Evaluate log 5 6252
Without a calculator:
Evaluate log10 10
Solution
1

Simplify 6252

“The log is the number I must raise the 5 to
in order to get _________.”
“The log is the number I must raise the 10 to in
order to get 10.”
**SO, BIG TAKE AWAY: When base
and number match, the log = 1.
Without a calculator:
Evaluate log 2 1
“The log is the number I must raise the 2 to in
order to get 1.”
**SO, BIG TAKE AWAY: When number
= 1, the log = 0. This is because
anything to the 0-power is 1.
*Log rules:
Remember- A logarithm is an ____________________. Similar laws apply to logs that hold for exponents.
When we multiply expressions of like bases we add exponents
𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
Using what we know about exponents:
Multiply…
When we divide expressions of like bases we subtract exponents
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
log 𝑥𝑦 =________________________
Divide…
When we raise an expression to a power we multiply exponents
(𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
Important facts to memorize…
𝑥
log 𝑦 =_____________________
Power…
log 𝑥 𝑛 = 𝑛 log 𝑥
*This is on page _______ in our formula booklets!
IB Math SL Year 2
Let’s try it!
Example
Express as a single logarithm:
log 5 + log 6
My thinking
“When adding logs with the same
base, we can multiply the big numbers
and take the log of that answer.”
Express as a single logarithm:

Simplify/condense the coefficient
as an exponent.

“When subracting logs with the
same base, we can divide the big
numbers and take the log of that
answer.”
Solution
3log x - 2log y
Station 2: Change of base
-
Sometimes you need to change the base of a logarithm and there is a formula that enables you to do this.
-
We most often find ourselves using the formula to change logarithms into base 10 so we can then use the log
button on our calculator.
*This is on page _______ in our formula booklets!
Let’s look at some:
Example
Use the change of base formula to
evaluate log 4 9 to 3 significant figures.
My thinking
“Here we have to use change of base
because our calculator can only do
logarithms of base 10.”
log 𝑥 3 = 𝑎 and log 𝑥 6 = 𝑏. Find
log 3 6 in terms of 𝑎 and 𝑏.

Use the change of base formula to
1
evaluate log 5 7

Evaluate each piece (replace with
a and b)
Use change of base formula.
Solution
IB Math SL Year 2
Station 3: SOLVING EQUATIONS
Let’s see what we already know how to do:
1. Without a calculator: Solve for x: 2𝑥 = 32
2. Without a calculator: Solve for the exact value of x: 2𝑥 = 30
*What is the problem we encounter? __________________________________________________________
Let’s Try Three Together…
1. Solve log 𝑎 (𝑥 2 ) = log 𝑎 (3𝑥 + 4)
2. Solve ln(12 − 𝑥) = ln 𝑥 + ln(𝑥 − 5)
IB Math SL Year 2
Station 4: Mixed Practice (to be completed and checked with answer key)
1. Write these expressions in log form.
a. X = 29
b. x = ab
2. Write these expressions in exponent form:
a. X = 𝑙𝑜𝑔2 8
b. X = 𝑙𝑜𝑔𝑎 𝑏
3. Evaluate:
a. log 2 25
b. log 64 4
1
81
𝒄. log 5 √5
d. log 3
4. Without a calculator:
a. log 6 6
b. log 8 1
5. Find the value of 𝑥 if log 2 𝑥 = 5
1
2
6. Express log 2 5 + log 2 36 − log 2 10 as a single logarithm.
IB Math SL Year 2
√𝑥
7. Exam Style: Given that 𝑎 = log 5 𝑥, 𝑏 = log 5 𝑦, and 𝑐 = log 5 𝑧, write log 5 (𝑦2 𝑧3 ) in terms of 𝑎, 𝑏, and 𝑐.
8. Rewrite as a single logarithm:
a. log24 – log2
b. log x - log y – log z
9. Solve:
a. log 4 𝑥 = 3
b. log 𝑥 64 = 2
10. Without a calculator:
a. log 𝑥 = 2
b. log x = -1
c. log x = 0
d. log 𝑛 𝑛
11. Use a calculator to
a. evaluate 𝐥𝐨𝐠 𝟐 to 3 decimal places.
b. Solve these equations, giving your answers to 3 significant figures.
a. log 𝑥 = 3
b. 10𝑥 = 0.75
12. Write these expression in the form a + blogx, where a and b are integers. In other words, rewrite each log expression
below using log rules.
IB Math SL Year 2
13.
14. Use the change of base formula to evaluate these expressions to 3 significant figures.
a. 𝑙𝑜𝑔2 7
b. 𝑙𝑜𝑔3 (0.7)
15. Solve for x: 8𝑥 − 5(4𝑥 ) = 0
IB Math SL Year 2
16. Solve these equations for x:
a. 𝑙𝑜𝑔9 (𝑥 − 2) = 2
b. 𝑙𝑜𝑔1 (3 − 𝑥) = 5
2
17. Given that 𝑙𝑜𝑔2 𝑥 + 𝑙𝑜𝑔2 (2𝑥 + 7) = 𝑙𝑜𝑔2 𝐴, find an expression for A in terms of x. Hence or otherwise solve
𝑙𝑜𝑔2 𝑥 + 𝑙𝑜𝑔2 (2𝑥 + 7) = 2.
Now, go back through your notes and
highlight the questions you need to go
over with a peer or in extra help!!