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Exponential Equations
Like Bases
Warm Up
The following quadratic equation has exactly one solution for x. Find the value of k. Explore more than one method…..
0,0944 2 kkxx
Method One
One solution indicates that the discriminant is equal to 0:
3
0sin3
03316
0916
014416
094416
04
2
2
2
2
k
kcebutk
kk
k
k
k
acb
Method Two
....0944 2 sosquareperfectaiskxx
solutiononeonlyisthereBecause
3
124
91243232
.329442
22
k
k
xxxx
inspectionbyxkxx
Warm Up - If $5,000 is invested at an interest rate of 5% per year, find the amount of the investment at the end of 5 years for the following compounding methods:
Annually
Monthly
Quarterly
Continuously
Annually: Monthly:
41.6381
1
05.015000
1
51
A
A
n
rPA
tn
79.6416
12
05.015000
1
512
A
A
n
rPA
tn
Quarterly: Continuously
19.6410
4
05.015000
1
54
A
A
n
rPA
tn
13.6420
5000 505.0
A
eA
PeA rt
82 23 xSolve
3
1
13
323
22 323
x
x
x
x
429 327 xxSolve
31
42273
33
3342273
4293
x
xx
xx
xx
11
11
104213
33
33104213
52273
x
x
xx
xx
xx
527 927 xxSolve
9275 6432 xxSolve
13
89
8913
54123525
22
2254123525
926755
x
x
xx
xx
xx
82
14
x
Solve
7
7
34
22
2234
341
x
x
x
x
x
279
126
x
Solve
12
7
712
3412
33
333412
3262
x
x
x
x
x
77 3225.0 xxSolve
19
35
3519
35514
22
22
22
1
324
1
35514
35572
757
2
77
x
x
xx
xx
xx
xx
xx
Logarithmic Functions
Logarithms
Used to find unknown exponents in exponential models
Define many measurement scales in the sciences such as the pH, decibel, and Richter scales.
For any positive base b, where b≠1: bx = y if and only if x = logby
Exponential Form
Logarithmic Form
310 1000 10log 1000 3
Write in logarithmic form.
35 125
3125log 5
Write in logarithmic form.
46 1296
41296log 6
Write in logarithmic form. 3 1
327
327
1log 3
Natural log
log lne x x
xe x ln
The Natural Logarithmic Function
The natural logarithmic function
is the inverse of the natural exponential function
)ln(log xydabbreviatexy e
.xey
Write in logarithmic form. 2 7.389...e
2389.7ln
2389.7log
asSame
e
Write in exponential form. 3log 81 4
8134
Write in exponential form. 8log 512 3
51283
Write in exponential form. ln(181.3) 5.2
3.181
2.53.181log
2.5
e
e
Write in exponential form. 3
1log 2
9
9
13 2
Find the value of x.4log 64x
3
44
6443
x
x
x
Find the value of x.2log 16x
4
22
1624
x
x
x
Find the value of x. 125log 5x
3
1
13
55
512513
x
x
x
x
Find the value of x. 5 log 32x
2
232
32
32
5
155
5
15
15
5
x
xorx
x
x
Find the value of x.34 log x
x
x
81
34
Find the value of x. 2log 1x
0
12
x
x
Find the value of x.1
log 92 x
81
9
9
2
2
2
1
2
1
x
x
x
Find the value of x. 73 log x
x
x
343
73
Find the value of x. 9log 3x
4
12
12
33
39
2
12
x
x
x
x
Find the value of x. 1ln xe
1
1
1log
1
x
ee
ee
xe
x
x
e
Solve ( ) ln 44f x x for x
...784.344ln44 f
Find to the nearest thousandth – using the calculator.
10( ) log 240f x x for x
380.2
240log 10
xf
xf
Find to the nearest thousandth – using the calculator.
10 85x
929.1
85log 10
x
x
Find to the nearest thousandth – using the calculator.
10 22x
342.1
22log 10
x
x
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