intro to logs
TRANSCRIPT
![Page 1: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/1.jpg)
Introduction To Logarithms
![Page 2: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/2.jpg)
Our first question then must be:
Our first question then must be:
What is a logarithm ?What is a logarithm ?
![Page 3: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/3.jpg)
Definition of Logarithm
Definition of Logarithm
Suppose b>0 and b≠1, there is a number ‘p’
such that:
Suppose b>0 and b≠1, there is a number ‘p’
such that:
logb n p if and only if bp n
![Page 4: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/4.jpg)
You must be able to convert an exponential
equation into logarithmic form and vice versa.
So let’s get a lot of practice with this !
![Page 5: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/5.jpg)
Example 1:
Solution: log2 8 3
We read this as: ”the log base 2 of 8 is equal
to 3”.
3Write 2 8 in logarithmic form.
![Page 6: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/6.jpg)
Example 1a:
Write 42 16 in logarithmic form.
Solution: log4 16 2
Read as: “the log base 4 of 16 is
equal to 2”.
![Page 7: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/7.jpg)
Example 1b:
Solution:
Write 2 3 1
8 in logarithmic form.
log2
1
8 3
1Read as: "the log base 2 of is equal to -3".
8
![Page 8: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/8.jpg)
Okay, so now it’s time for you to try some on
your own.
1. Write 72 49 in logarithmic form.
7Solution: log 49 2
![Page 9: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/9.jpg)
log5 10Solution:
2. Write 50 1 in logarithmic form.
![Page 10: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/10.jpg)
3. Write 10 2 1
100 in logarithmic form.
Solution: log10
1
100 2
![Page 11: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/11.jpg)
Solution: log16 4 1
2
4. Finally, write 161
2 4
in logarithmic form.
![Page 12: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/12.jpg)
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.
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.
![Page 13: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/13.jpg)
Example 1:
Write log3 814 in exponential form
Solution: 34 81
![Page 14: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/14.jpg)
Example 2:
Write log2
1
8 3 in exponential form.
Solution: 2 3 1
8
![Page 15: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/15.jpg)
Okay, now you try these next three.
1. Write log10 100 2 in exponential form.
3. Write log27 3 1
3 in exponential form.
2. Write log5
1
125 3 in exponential form.
![Page 16: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/16.jpg)
1. Write log10 100 2 in exponential form.
Solution: 102 100
![Page 17: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/17.jpg)
3. Write log27 3 1
3 in exponential form.
Solution: 271
3 3
![Page 18: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/18.jpg)
When working with logarithms,if ever you get “stuck”, tryrewriting the problem in
exponential form.
When working with logarithms,if ever you get “stuck”, tryrewriting the problem in
exponential form.
Conversely, when workingwith exponential expressions,
if ever you get “stuck”, tryrewriting the problemin logarithmic form.
Conversely, when workingwith exponential expressions,
if ever you get “stuck”, tryrewriting the problemin logarithmic form.
![Page 19: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/19.jpg)
Solution:Let’s rewrite the
problem in exponential form.
62 x
We’re finished !
6Solve for x: log 2x
Example 1
![Page 20: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/20.jpg)
Solution:
5y 1
25
Rewrite the problem in exponential form.
Since 1
255 2
5y 5 2
y 2
5
1Solve for y: log
25y
Example 2
![Page 21: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/21.jpg)
Example 3
Evaluate log3 27.
Try setting this up like this:
Solution:
log3 27 y Now rewrite in exponential form.
3y 273y 33
y 3
![Page 22: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/22.jpg)
Properties of logarithms
![Page 23: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/23.jpg)
Let b, u, and v be positive numbers such that b≠1.
Product property:
logbuv = logbu + logbvQuotient property:
logbu/v = logbu – logbvPower property:
logbun = n logbu
![Page 24: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/24.jpg)
Expanding Logarithms
You can use the properties to expand logarithms.
log2 7x3 / y= log27x3 - log2y =
log27 + log2x3 – log2y =
log27 + 3·log2x – log2y
![Page 25: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/25.jpg)
Expand:
log 5mn =
log 5 + log m + log nExpand:
log58x3 =
log58 + 3·log5x
![Page 26: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/26.jpg)
Condensing Logarithms
log 6 + 2 log2 – log 3 =
log 6 + log 22 – log 3 =
log (6·22) – log 3 = log 24 – log 3=
log 24/3= log 8
![Page 27: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/27.jpg)
Condense:
log57 + 3·log5t = log57t3
Condense:
3log2x – (log24 + log2y)= log2
x3/4y
![Page 28: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/28.jpg)
Change of base formula:u, b, and c are positive numbers with b≠1 and
c≠1.
Then:
logcu = log u / log c (base 10)
![Page 29: Intro to Logs](https://reader033.vdocument.in/reader033/viewer/2022052620/5577dee1d8b42a7b7b8b4a1e/html5/thumbnails/29.jpg)
Examples:
Use the change of base to evaluate:
log37 =
log 7 ≈ 1.771
log 3