Sec 5.5
Properties of Logarithms
Math 1051 - Precalculus I
Properties of Logarithms
Sec 5.5
Sec 5.5 Properties of Logarithms
Solve:
1 + log(x + 2) = 2
Properties of Logarithms
Sec 5.5
Sec 5.5 Properties of Logarithms
Solve:
1 + log(x + 2) = 2
Ans: x = 8
Properties of Logarithms
Sec 5.5
Properties of Logs
Properties of Logarithms
Sec 5.5
Properties of Logs
loga ar = r
Properties of Logarithms
Sec 5.5
Properties of Logs
loga ar = r
aloga M = M
Properties of Logarithms
Sec 5.5
Properties of Logs
loga ar = r
aloga M = M
loga 1 = loga a0 = 0
Properties of Logarithms
Sec 5.5
Properties of Logs
loga ar = r
aloga M = M
loga 1 = loga a0 = 0
loga a = loga a1 = 1
Properties of Logarithms
Sec 5.5
Properties of Logs
loga ar = r
aloga M = M
loga 1 = loga a0 = 0
loga a = loga a1 = 1
loga MN = loga M + loga N
Properties of Logarithms
Sec 5.5
Properties of Logs
loga ar = r
aloga M = M
loga 1 = loga a0 = 0
loga a = loga a1 = 1
loga MN = loga M + loga N
loga
M
N
= loga M − loga N
Properties of Logarithms
Sec 5.5
Properties of Logs
loga ar = r
aloga M = M
loga 1 = loga a0 = 0
loga a = loga a1 = 1
loga MN = loga M + loga N
loga
M
N
loga
Mr
= loga M − loga N
= r loga M
Properties of Logarithms
Sec 5.5
A Little Bit of History
Back in the day people had to divide numbers by hand using
long division:
Properties of Logarithms
Sec 5.5
A Little Bit of History
Back in the day people had to divide numbers by hand using
long division:
123000
29
Properties of Logarithms
Sec 5.5
A Little Bit of History
Back in the day people had to divide numbers by hand using
long division:
123000
29
By using logarithms and their properties people could turn
“multiplication” into “addition”
Properties of Logarithms
Sec 5.5
A Little Bit of History
Properties of Logarithms
Sec 5.5
A Little Bit of History
Properties of Logarithms
Sec 5.5
A Little Bit of History
Properties of Logarithms
Sec 5.5
A Little Bit of History
Thankfully, today we have calculators
Properties of Logarithms
Sec 5.5
Remember the rule:
If aM = aN then M = N
Properties of Logarithms
Sec 5.5
Remember the rule:
If aM = aN then M = N
We used this to solve something like: 23x−8 = 16.
Properties of Logarithms
Sec 5.5
Remember the rule:
If aM = aN then M = N
We used this to solve something like: 23x−8 = 16.
We get similar rules for logarithms:
Properties of Logarithms
Sec 5.5
Remember the rule:
If aM = aN then M = N
We used this to solve something like: 23x−8 = 16.
We get similar rules for logarithms:
If loga M = loga N then M = N
Properties of Logarithms
Sec 5.5
Remember the rule:
If aM = aN then M = N
We used this to solve something like: 23x−8 = 16.
We get similar rules for logarithms:
If loga M = loga N then M = N
If M = N then loga M = loga N
Properties of Logarithms
Sec 5.5
Change of Base Formula
Suppose you have y = loga M but you don’t like the base a.
Properties of Logarithms
Sec 5.5
Change of Base Formula
Suppose you have y = loga M but you don’t like the base a.
You can “change” into another base using the change of base
formula:
Properties of Logarithms
Sec 5.5
Change of Base Formula
Suppose you have y = loga M but you don’t like the base a.
You can “change” into another base using the change of base
formula:
loga M =
Properties of Logarithms
logb M
logb a
Sec 5.5
Change of Base Formula
Suppose you have y = loga M but you don’t like the base a.
You can “change” into another base using the change of base
formula:
loga M =
logb M
logb a
This is an important rule to remember. You should either
memorize it , or go through enough problems where you use it
to where the formula is second nature .
Properties of Logarithms
Sec 5.5
Change of Base Formula
Suppose you have y = loga M but you don’t like the base a.
You can “change” into another base using the change of base
formula:
loga M =
logb M
logb a
This is an important rule to remember. You should either
memorize it (ick), or go through enough problems where you
use it to where the formula is second nature (that way you don’t
have to memorize it!).
Properties of Logarithms
Sec 5.5
Another reason it might be useful to know the change of base
formula:
Properties of Logarithms
Sec 5.5
Another reason it might be useful to know the change of base
formula:
Properties of Logarithms
Sec 5.5
Examples
Find y = log4 16
Properties of Logarithms
Sec 5.5
Examples
Find y = log4 16
Simplify eloge2 9
Properties of Logarithms
Sec 5.5
Examples
Find y = log4 16
Simplify eloge2 9
Simplify log3 8 · log8 9
Properties of Logarithms
Sec 5.5
Use properties to rewrite logs:
Write as a sum/difference of logs:
3√
x x +1
log
(x − 2)2
Properties of Logarithms
Sec 5.5
Use properties to rewrite logs:
Write as a sum/difference of logs:
3√
x x +1
log
(x − 2)2
Write as a single log:
1
1
log(x 3 + 1) + log(x 2 + 1)
3
2
Properties of Logarithms
Sec 5.5
Use properties to rewrite logs:
Write as a sum/difference of logs:
3√
x x +1
log
(x − 2)2
Write as a single log:
1
1
log(x 3 + 1) + log(x 2 + 1)
3
2
Write as a single log:
3 log2 (x) + log4 (x − 2)
Properties of Logarithms
Sec 5.5
Read section 5.6 for next Monday!
No class this Wednesday.
Properties of Logarithms
Sec 5.5