7/1/2015
Properties of Logarithms
 Recall:
Ch. 3 – Exponential and Logarithmic
Functions
 ax = ay means x = y
 loga ax = x
3.4 – Solving Exponential and Logarithmic
Equations
Equivalent forms
 loga x = loga y means x = y
 aloga x = x
Inverse properties
 Strategies for solving these equations:
1.
2.
 Solve: 2x = 32
 Rewrite the 32 into an exponential phrase!
 2x = 25  x = 5
 Solve: 3(2x) = 42
 Solve for x just like always… get x by itself!
 2x = 14
 log2 2x = log2 14
 x = log2 14
 Use change of base to get x = 3.807
 Solve: log x – log 12 = 0
 log x = log 12  x = 12
 Solve: 6e2x – 10 = 2
 Get the e by itself first…
 6e2x = 12
 e2x = 2
 ln e2x = ln 2
 2x = ln 2
 x = (ln 2)/2  x = .347
 Solve: ln x = -3
 Use inverse properties!
 eln x = e-3
 x = e-3  x = .0498
 These are the easy ones! They get harder!
Solve for x:
ln x  2.4
log5 (9 x  6)  log5 (4 x  8)
3. .883
3. 90.597
4. .380
4. .640
5. 6.52
5. 1.563
0%
0%
6.
52
0%
.3
8
0%
.8
83
0%
11
.0
2
.8
75
0%
0%
0%
0%
0%
1.
56
3
2. .357
.6
4
2. 11.02
90
.5
97
1. 2.8
.3
57
1. .875
2.
8
Solve for x:
Rewrite the equation in equivalent exponential/log form
Use inverse properties to cancel out logs and exponent bases
1
7/1/2015
Solve for x.
Round to the
nearest tenth.
log 2 (10 x  5)  6
Solve for x:
24 x  2  8
1. .25
2. .5
1. 1.7
3. -.274
2. 4.1
4. 1.5
3. 6.9
5. 15.5
2
3x
0%
0%
1.
5
.2
5
0%
15
.5
0%
.5
0%
0%
.8
0%
.6
0%
6.
9
1.
7
Solve for x. Round to the nearest
tenth.
 565
log 3 ( x  9)  log 3 (4 x  7)
3. -3.3
3. -1.8
4. 0.2
4. 2.4
5. -6.1
0%
0%
0%
0%
0%
5. No solution
 Solve: e2x – 3ex + 2 = 0
 It’s like a quadratic, but with e’s, so write it as a quadratic!
 (ex)2 – 3ex + 2 = 0
 Factor…
 (ex – 2)(ex – 1) = 0
 Break into 2 equations!
ex  2  0
ex  2
e x 1  0
ex  1
x  ln 1
x0
x  ln 2
x  0.693
x  0, 0.693
0%
0%
0%
0%
so
lu
No
-6
.1
.2
-3
.3
2.
7
1.
8
-5
.3
0%
n
2. 0.7
tio
2. 2.7
2.
4
1. -5.3
-1
.8
1. 1.8
.7
Solve for x.
Round to the
nearest tenth.
0%
4.
1
0%
5. 0.8
-.2
74
4. 0.6
 Solve: ln (x – 2) + ln (2x – 3) = 2 ln x
 We can’t get rid of the ln’s until there are only 1 per side of the
equation…
 …so we use our addition and exponent property of logs!
 ln ((x – 2)(2x – 3)) = ln x2
 Now we can lose the ln’s!
 (x – 2)(2x – 3) = x2
 2x2 – 7x + 6 = x2
 x2 – 7x + 6 = 0
 Factor…
 (x – 6)(x – 1) = 0
 x = 1, 6
 WAIT!!! CHECKYOUR SOLUTIONS!!!
 x = 1 is extraneous, so final answer is x = 6
2
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Write the exponential in logarithmic
form.
Solve for x. Round to the nearest
hundredth.
1. log3 4 = 64
1. -1
2. log3 64 = 4
2. 5.39
3. log4 3 = 64
3. 0.52
ln x  2  1
43  64
4. 0
0%
0%
So
64
lu
t io
n
0
0%
.5
2
-1
3
0%
No
lo
g4
lo
g6
4
lo
g4
0%
5. No Solution
=
3
0%
=
64
0%
4
=
3
64
=
4
lo
g3
lo
g3
0%
=
64
5. log4 64 = 3
0%
4
0%
5.
39
4. log64 4 = 3
Solve for x. Round to the nearest
tenth.
log 2 x  log 2 ( x  8)  log 2 9
Solve for x. Round to the nearest
hundredth.
12
1. 9
1. 0.08
2. 8.5
2. -0.51
3. -1, 9
3. 0.76
4. -0.88
0%
-.8
8
0%
So
lu
t io
n
0%
.7
6
0%
No
No
So
0%
5. No Solution
.0
8
0%
4.
1
0%
lu
t io
n
0%
-1
,9
9
5. No Solution
0%
8.
5
0%
-.5
1
4. 4.1
9
3  4e x
Math Team Problem: Solve for w…
1
1
(log k 64)  (log k 16)  log k w
3
4
1. 25.3
2. 6
3. 8
4. 16
0%
0%
85
.3
0%
16
0%
8
25
.3
0%
6
5. 85.3
3