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…..
4 x  4kx  9  0, k  0
2
Method One
 One solution indicates that the
discriminant is equal to 0:
b 2  4ac  0
16k 2  44 9   0
16k 2  144  0


16 k 2  9  0
16k  3k  3  0
k  3 but sin ce k  0
k 3
Method Two
Because there is only one solution
4 x 2  4kx  9  0 is a perfect square so....
4 x  4kx  9  2 x  3 by inspection.
2
2
2 x  32 x  3  4 x 2  12 x  9
4k  12
k 3
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:

A  P 1 

r

n
 Monthly:
nt
 0.05 
A  50001 

1 

A  6381.41

A  P 1 

15
r

n
nt
 0.05 
A  50001 

12 

A  6416.79
125
 Quarterly:
 r
A  P 1  
 n
 Continuously
nt
 0.05 
A  50001 

4 

A  6410.19
A  Pe
45
rt
0.055
A  5000e
A  6420.13
Solve 2
3x2
2
8
3x2
2
3
3x  2  3
3x  1
1
x
3
Solve 27
3
x 9
3 x  9 
3 x  27
3
3
2 x 4
2 x4
2 x4
3
3
3 x  27  2 x  4
x  31
Solve 27
3 x  7 
3
x 7
3
9
2  2 x 5 
33 x  21  34 x 10
3 x  21  4 x  10
 x  11
x  11
2 x 5
Solve 32
2
2
5 x 7
5 5 x  7 
25 x  35
2
 64
2 x 9
6  2 x 9 
12 x 54
2
25 x  35  12 x  54
13 x  89
89
x
13
1
Solve  
2
x4
8
2
1 x  4 
 x4
2
2
2
x43
x7
x  7
3
3
1
Solve  
9
6 x2
 27
3
2 6 x  2 
12 x  4
3
3
3
3
 12 x  4  3
 12 x  7
7
x
12
3
Solve 0.25
7x
1
 
4
 32
7x
 1 
 2
2 
2 
 32 x 7
7x
2 7 x
 2 5 x 7 
 2 5 x 35
2 14 x  2 5 x 35
 14 x  5 x  35
 19 x  35
35
x
19
x 7
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
10  1000
3
Logarithmic
Form
log10 1000  3
Write in logarithmic
form.
5  125
3
log 5 125  3
Write in logarithmic
form.
6  1296
4
log 6 1296  4
Write in logarithmic
form.
1
3 
27
3
1
log 3
 3
27
Natural log
loge x  ln x
e
ln x
x
The Natural Logarithmic
Function
 The natural logarithmic function
y  log e x (abbreviated y  ln x)
is the inverse of the natural
exponential function
ye .
x
Write in logarithmic
form.
2
e  7.389...
log e 7.389  2
Same as
ln 7.389  2
Write in exponential form.
log3 81  4
3  81
4
Write in exponential form.
log8 512  3
8  512
3
Write in exponential form.
ln(181.3)  5.2
log e 181.3  5.2
e
5.2
 181.3
Write in exponential form.
3
2
1
log 3  2
9
1

9
Find the value of x.
x  log 4 64
4  64
x
4 4
x3
x
3
Find the value of x.
x  log 2 16
2  16
x
2 2
x4
x
4
x  log125 5
Find the value of x.
125  5
x
3x
5 5
3x  1
1
x
3
1
Find the value of x.
5  log x 32
x  32
5
x   32
1
5 5
1
5
 
x  32 or x  2
5
x2
1
5 5
4  log3 x
Find the value of x.
3 x
81  x
4
x  log 2 1
Find the value of x.
2 1
x0
x
1
 log x 9
2
Find the value of x.
1
2
x 9
2
 
2
x   9
 
 
x  81
1
2
3  log7 x
Find the value of x.
7 x
343  x
3
x  log9 3
Find the value of x.
9  3
x
3
1
2x 
2
1
x
4
3
2x
1
2
Find the value of x.
1
ln    x
e
1
log e  x
e
1
x
e 
e
x
1
e e
x  1
Solve
f ( x)  ln x
for
x  44
f 44  ln 44  3.784...
Find to the nearest thousandth – using the calculator.
f ( x)  log10 x
for
x  240
f  x   log 10 240
f  x   2.380
Find to the nearest thousandth – using the calculator.
10  85
x
log 10 85  x
x  1.929
Find to the nearest thousandth – using the calculator.
10  22
x
log 10 22  x
x  1.342