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Logs - Day 1
TUESDAY, FEBRUARY 21, 2017
DEFINE THE TERMS ON PAGE 291 INTO YOUR VOCAB LIST!
ADAPTED FROM:
HT TP://MATHEQUALSLOVE.BLOGSPOT.COM/2014/01/INTRODUCINGLOGARITHMS-WITH-FOLDABLES.HTML
Essential Question
How do I convert an exponential equation to a logarithmic equation
and vice versa?
How do I evaluate common logs and natural logs with the calculator?
What is a logarithm?
A logarithm is just a special way to ask a specific question.
The Question: What exponent is required to go from a base of b to
reach a value of a?
“log base b of a is x”
Exponential Form
exponent
base
= answer
Logarithmic Form
logbaseanswer= exponent
Anytime the base of a logarithm is not
written, it is assumed to be the number 10.
E
The logarithm answers the
question: What power do I raise
the base to in order to get _____?
The Logarithm Loop Trick
(for changing forms)
Always draw your loop counter-clockwise from the base!
Examples:
1. log 4 64 = 𝑥𝑥 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
4𝑥𝑥 = 64
2. 343𝑥𝑥 = 7 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
log 343 7 = 𝑥𝑥
log form
Let’s Practice!
HINTS: 1) THE BASE OF THE LOGARITHMIC FORM IS ALSO THE BASE OF
THE EXPONENTIAL FORM…
2) REMEMBER, A LOGARITHM IS ALWAYS EQUAL TO THE POWER OF THE
BASE TO GET THAT NUMBER!
Example 1:
Write 2 = 8 in logarithmic form.
3
Solution:
log2 8 = 3
We read this as: ”the log base 2 of 8 is equal to 3”.
Example 1a:
2
Write 4 = 16 in log arithmic form.
Solution:
log4 16 = 2
Read as: “the log base 4 of 16 is equal to 2”.
NOTE: Logarithms can be equal to negative numbers BUT…
The base of a logarithm cannot be negative…
NOR can we take the log of a negative number!!
Example 1b:
Write 2
Solution:
−3
1
in log arithmic form.
=
8
1
log2 = − 3
8
1
Read as: "the log base 2 of is equal to -3".
8
Okay, so now it’s time for
you to try some on your
own.
1. Write 72 = 49 in log arithmic form.
Solution: log 7 49 = 2
2. Write 5 = 1 in log arithmic form.
0
Solution:
log5 1 = 0
This brings up a special property of logarithms. Since anything to
the power of 0 gives us 1, what will the log (base anything) of 1
always give us?
logx 1 = 0
−2
3. Write 10
Solution:
1
=
in log arithmic form.
100
1
log10
= −2
100
1
2
4. Finally, write 16 = 4
in log arithmic form.
Solution:
1
log16 4 =
2
Do you remember what a half-power
represents?
A half-power is a SQUARE ROOT.
It is also very important to be able to start with a logarithmic
expression and change this into exponential form.
This is simply the reverse of
what we just did.
Example 1:
Write log3 81 = 4 in exp onential form
Solution:
3 = 81
4
Example 2:
1
Write log2 = − 3 in exp onential form.
8
Solution:
−3
2
1
=
8
Okay, now you try these next three.
1. Write log10 100 = 2 in exp onential form.
Solution: 10 = 100
2
1
2. Write log5
= − 3 in exp onential form.
125
Solution:
1
5 =
125
−3
1
3. Write log27 3 =
in exp onential form.
3
Solution:
1
3
27 = 3
Evaluating logarithms
HINTS: 1) USE THE LOG BUT TON (BASE 10) OR THE LN BUT TON (BASE
“E”) DEPENDING ON THE BASE.
2) FOR OTHER BASES, YOU WILL NEED TO KNOW THE “CHANGE-OF-BASE”
FORMULA WHICH WE WILL LEARN LATER.
Evaluate the following to 4 decimal places.
1) log 76
2) log 0.43
… 1.8808
3) ln 76
… –0.3665
4) ln 0.43
… 4.3307
5) log 56
… –0.8440
6) log .76
… 1.7482
… –.1192
Intro to Logs Practice:
Write the form given. Identify it as “Log Form” or “Exponential Form” and then convert it to
the other form.
1) 5x = 56
2) logx+1 (34) = 13
3) log7 (3x – 2) = 128
4) w(x – 3) = 6y
Use your calculator to evaluate the following log values to 4 decimal places.
5) log 54
6) log 0.64
7)
8) log 1.45
log10 87
9) ln 78
10) ln 1.26
Intro to Logs Practice: KEY
Write the form given. Identify it as “Log Form” or “Exponential Form” and then convert it to
the other form.
1) 5x = 56
log5 (56) = x
3) log7 (3x – 2) = 128
7128 = 3x – 2
2) logx+1 (34) = 13
(x + 1)13 = 34
4) w(x – 3) = 6y
logw (6y) = x – 3
Use your calculator to evaluate the following log values to 4 decimal places.
5) log 54
1.7324
7)
log10 87
1.9395
9) ln 78
4.3567
6) log 0.64
– 0.1938
8) log 1.45
0.1614
10) ln 1.26
0.2311