Section 5.2 Logarithms (Page 437)
Let’s say we want to solve this equation 2x = 8. We can guess the answer.
What about 2 x = 16? (But wait, What about 2x = 12. How we get that? (We
could plot 2x and look on the curve)
or we could . . .
Use the calculator to raise 2 to powers between 3 and 4 until we guessed the
right response. Let’s do that!!!!
What about 10x = 100,
10x = 1000,
10x = 10,000
10x = 500?
Now find Log (500) on calculator. What is the answer???? No, what is the
answer?? IT’S THE EXPONENT YOU HAVE TO RAISE the base 10 to get the
Number!
Try it. 10 (2.6989) = ???????? (499.99)
Define a logarithm: “A logarithm is just an exponent of some base.”
More explicitly if bx = N the x is the power to which “a” is raised to get N.
So
x = logarithmb N or x = logb N
So if:
bx = N
then
x = logb N
Again: If 102.5 = 316.23, then log10 316.23 = 2.5
“The logarithm of 316.23,base 10 is 2.5, means that “2.5 ” is the
power you raise the base “10 ” to get 316.23.”
bx = N
↔ x = logb N
Let’s use this identity to solve a problem:
10x =242
So if:
102x = 220
x = logb N
Solve x = Log2 8
↔
101/2 x = 348
bx = N
(not on your calculator, is it?)
What is the log2 ¼
?
Oh, almost forgot log10N is the same as log N
Let’s rewrite and evaluate these:
#1. log3 9
#2. log2 64
#4 log2 √2
#5 log5 1
#3. Log √10
#6 Log10 10
Convert these to Logarithmic Equations
1
#7: 5 -3 =
#8: 10 0.3010 = 2
125
#9: Qt = x
#10 e-1 = 0.3679
Convert to exponential equation:
#11: t = log47
#12: log 7 = 0.845
#13 : logt Q = k
Oh yes, another thingy, there is a number “e” that is also the base of a
logarithm system:
For example loge 5 is shown as ln 5. So, . . .
ex = N is the same as x = ln (N) It’s called the “Natural Log System.”
So what is the ln (2) = ___________
Then, is e 0.6931 = 2 ???
Find these logarithms on calculator:
#14 : ln (50)
#15: ln 0.00037
#16: log 93,100
Let’s do some other things:
Log 1 = __________________ ln 1 = __________________
Write these logs above as exponential equations;
_________________________
If ,
log 101 = 1 ,
___________________________
log 102 = 2,
log 103 = 3
Then: log 10a =__________, and log 102r = ______________
What about the “e” system?
Ln e = _____________ ln e2 = ___________ and ln e3 = __________
What about ln ex _________ and ln e2x_____________
Find ln 10 = _________
log e = ________
Using your calculator, Let’s find these logarithms:
Log10 10 =
Log10 100 =
Log10 1000 =
Log10 20 =
Log10 200 =
Log10 2000 =
Log10 30 =
Log10 300 =
Log10 3000 =
Log10 40 =
Log10 400 =
Log10 4000 =
Log10 50 =
Log10 500 =
Log10 5000 =
Log10 60 =
Log10 600 =
Log10 6000 =
Log10 70 =
Log10 700 =
Log10 7000 =
Log10 80 =
Log10 800 =
Log10 8000 =
Log10 90 =
Log10 900 =
Log10 9000 =
Log10 100 =
Log10 1000=
Log10 10000 =
Log10 120 =
Log10 1200 =
Log10 12000 =
log10 0.01=
log10 0.1=
log10 0.001=
log10
log10
𝟏
𝟏𝟎𝟎𝟎
𝟏
𝟏𝟎𝟑
=
=
log10 𝟏𝟎−𝟑 =
log10
log10
𝟏
𝟏𝟎𝟎
𝟏
𝟏𝟎𝟐
=
=
log10 𝟏𝟎−𝟐 =
log10
log10
𝟏
𝟏𝟎
𝟏
𝟏𝟎𝟏
=
=
log10 𝟏𝟎−𝟏 =
__________________________________________________________________
Log10 10 = 1.000 , Then 101 = 10
Log10 20 = 1.3010 , Then 10(1.3010) =
Log10 30 =
Log10 40 =
Log10 50 =
Log10 60 =
Log10 70 =
Log10 80 =
Log10 90 =
Log10 100 =
Log10 120 =