Notes - Logarithmic Properties Def: Logarithms are inverses of

Notes - Logarithmic Properties
Def:
Logarithms are inverses of exponential functions.
y = logbx
y = bx
Therefore, y = logbx is equivalent to x = by
logarithmic form
Ex1
exponential form
Write each equation in exponential form:
a) 2 = log5x
b) 3 = logb64
y = logbx → by = x
y = logbx → by = x
52 = x
Ex2
b3 = 64
c) log37 = y
y = logbx → by = x
d) logba = r
y = logbx → by = x
3y = 7
br = a
Express each in logarithmic form:
a) 122 = x
by = x → y = logbx
log12x = 2
b) b3 = 8
by = x → y = logbx
logb8 = 3
c) ey = 9
by = x → y = logbx
d) rm = v
by = x → y = logbx
loge9 = y
logrv = m
On a separate sheet of paper to turn in at the end of class:
Express each in exponential form:
Express each in exponential form:
1) 4 = log216
1) 4 = log216 → 24 = 16
2) 2 = log9x
2) 2 = log9x → 92 = x
3) 3 = logb27
3) 3 = logb27 → b3 = 27
4) log5125 = y
4) log5125 = y → 5y = 125
Express each in logarithmic form:
Express each in logarithmic form:
5) 23 = 8
5) 23 = 8 → log28 = 3
2
6) 15 = x
6) 152 = x → log15x = 2
7) b3 = 1000
7) b3 = 1000 → logb1000 = 3
y
8) 300 = 8
8) 300 = 8y → log8300 = y
Properties of Logarithms
In general, with any base b:
logbb = 1
logb1 = 0
logbbx = x
blog x = x
Def:
Natural Log :
lne = 1
ln1 = 0
lnex = x
elnx = x
Common Log:
log10 = 1
log1 = 0
log10x = x
10logx = x
The common log is log base 10. We don't write the base 10 (log10), we simply write log. It is
understood.
On a separate sheet of paper to turn in at the end of class:
Simplify each using the
15) 7log723
Simplify each using the
properties of logarithms.
16) log108
properties of logarithms.
log33
9) log1111
17) 10
9) log1111 = 1
10) log41
18) ln1
10) log41 = 0
11) log61
19) lne
11) log61 = 0
7
6
12) log55
20) lne
12) log557 = 7
6
ln125
13) log44
21) e
13) log446 = 6
log 19
9x
14) 8 8
22) lne
14) 8log819 = 19
15) 7log723 = 23
16) log108 = 8
17) 10log33 = 33
18) ln1 = 0
19) lne = 1
20) lne6 = 6
21) eln125 = 125
22) lne9x = 9x
What if you are asked to simplify a logarithm that is not expressed as a property?
Ex3
Evaluate log39
think 3 to the what power is 9? → well, 32 = 9, so log39 = 2
Evaluate log5125
think 5 to the what power is 125? → well, 53 = 125, so log5125 = 3
Evaluate log164
think 16 to the what power is 4? → well, 16 = 4, and 16 = 16!/! , so 16!/! = 4
therefore, log164 = ½
On a separate sheet of paper to turn in at the end of class:
Evaluate each:
Evaluate each:
23) log216
23) log216 = 4
24) log100
24) log100 = 2
25) log366
25) log366 = 1/2
26) log749
26) log749 = 2
27) log327
27) log327 = 3
28) log264
28) log264 = 6
Ex4
Evaluate log5140
think 5 to the what power is 140? → well, 53 = 125 and 54 = 625, hmm… so the answer must be
a decimal before 3 and 4, but closer to 3. Exactly what decimal?
The Change of Base Property
!"# (!)
!" (!)
𝑙𝑜𝑔! 𝑀 = !"# (!) 𝑎𝑛𝑑 𝑙𝑜𝑔! 𝑀 = !" (!)
Ex4
Evaluate log5140
I know that the answer must be between 3 and 4. Use the change of base formula:
!"# (!"#)
𝑙𝑜𝑔! 140 = !"# (!) = 3.07
Ex5
Evaluate log753
!" (!")
𝑙𝑜𝑔! 53 = !" (!) = 2.04

Download Report