Solving with Unlike
Bases

Warm Ups on the next 3 slides….
WARM UP
Solve.
3 x 1
32
1
 
4
x 5
2 
5 3 x 1
 2 2 
x 5
215 x 5  2  2 x 10
15 x  5  2 x  10
17 x  15
15
x
17
Graph.
Condense.
y  2x4  3
3log 2 x  4log 2 y  log 2 5
log 2 x 3  log 2 y 4  log 2 5
x3 y 4
log 2
5
Warm Up
Solve : log 4 log 3 x   0
4  log 3 x
0
log 3 x  1
3 x
x3
1
Warm Up
Solve : log 1 8  x
2
x
1
8
2
x
3
2 2
x3
x  3
The Common Logarithm


The base-10 logarithm is called the common
logarithm.
The common logarithm is usually written as
logx.
log 10 x  log x

If we can manage to make everything into log
base 10, we can use our calculators to solve.
For Example….Solve 5  62.
x
5  62
x
log 5  log 62
x log 5  log 62
log 62
x
log 5
x  2.56
x
Solve:
5  28
x
5  28
x
log 5 x  log 28
x log 5  log 28
log 28
x
log 5
x  2.07
Solve:
6.4  59
x
log 6.4  log 59
x
x log 6.4  log 59
log 59
x
log 6.4
x  2.20
Solve:
8  73
x
log 8  log 73
x
x log 8  log 73
log 73
x
log 8
x  2.06
Solve:
4  45
3x
log 4  log 45
3 x log 4  log 45
3x
log 45
3x 
log 4
3 x  2.74593
x  0.92
Solve:
5  29
x
log 5
x
 log 29
 x log 5  log 29
log 29
x
log 5
x  2.09
Solve:
x7
3
 76
x7
log 3  log 76
( x  7) log 3  log 76
log 76
x7 
log 3
log 76
x
7
log 3
x  3.06
Solve:
7  2  19
x
2  12
x
log 2  log 12
x log 2  log 12
log 12
x
log 2
x  3.58
x
Solve:
1
 
3
2 x
1
log  
3
 12
2 x
 log 12
1
 2 x log    log 12
3
log 12
 2x 
1
log  
3
 2 x  2.26186
x  1.13
Change of Base

There is a way to write equivalent logarithmic
expressions with different bases.

There is a formula to find the change of base.

Let’s derive the change of base formula…..
Let’s derive the change of Base Formula!
log b x
log b x  y
by  x
log b  log x
y log b  log x
y
log x
y
log b
log x
log b x 
log b
Change of Base Formula:
log x
log b x 
log b
Evaluate to the nearest hundredth.
log 6 78
log 78
log 6 78 
 2.43
log 6
Evaluate to the nearest hundredth.
log 2 23
log 23
log 2 23 
 4.52
log 2
Evaluate to the nearest hundredth.
log 1 35
4
log 35
 2.56
1
log
4
Evaluate to the nearest hundredth.
2  log 4 17
log 17
2
 4.04
log 4
Natural Log
Warm Up: Find the final amount of money if
you invested
$6700 compounded monthly at 5.4% for 20 years.
 r
A  P 1  
 n
nt
 0.054 
A  67001 

12 

A  19,681.61
1220 
$4500 compounded continuously at 7% for 20 years.
A  Pe
rt
A  4500e 0.0720
A  18,248.40
Warm Up: $1000 is invested at 15% per annum interest,
compounded monthly. Calculate the minimum number of
months required for the value of the investment to exceed
$3000.
 r
A  P 1  
 n
nt
12t
 0.15 
3000  10001 

12


3  1.0125
12t
log 3  log 1.0125
log 3  12t log 1.0125
log 3
12t 
log 1.0125
t  7.369762662 YEARS
t x 12  88.4  89 MONTHS
12t
Warm Up:
Let f x   x , and g x   2 x. Solve the equation
f
1

 g x   0.25
1st find f
1
y x
x
f
2 
x 2
2
2x
2
2x
y
y  x2
1
x  :
Next find the composition :
x   x
2
 0.25
1

4
2
2
2 x  2
x  1
Natural Base

e = 2.718281828…

Models a variety of situations of growth &
decay in nature

Represented as: ln
(means loge)
ln  log e
Evaluate to the nearest thousandth.
e
4
54.598
Evaluate to the nearest thousandth.
e
2.3
0.100
Evaluate to the nearest thousandth.
e
2
5
1.492
Evaluate to the nearest thousandth. (Think of 2
Ways…….
ln 6
With Calculator Directly :
ln 6  1.792
Use Change of Base 1st :
ln 6
 log e 6
log 6

log e
 1.792
Evaluate to the nearest thousandth.
ln(2.5)
0.916
Evaluate to the nearest thousandth.
ln(2)
Negative domain value!
No Solution
REMEMBER : Domain is
all POSITIVE real numbers.
Find
ln(e) 
log e e   x
e e
x 1
x
Simplify each expression.
e
ln 7
e x
log e x  ln 7
ln 7
ln x  ln 7
x7
Simplify each expression.
e
ln12
e x
log e x  ln 12
ln12
ln x  ln 12
x  12
Simplify each expression.
e
2ln3
e
x
log e x  2 ln 3
2 ln 3
ln x  ln 3
ln x  ln 9
x9
2
Simplify each expression. (2 Ways)
ln e
9
ln e  x
ln e  x
9
log e e  x
9
e e
x9
x
9
9
or
9 ln e  x
91  x
9x
Simplify each expression. (2 Ways)
6ln e
4
4  6ln e  x
6 ln e  x
4
6 log e e  x
4

6e e
x
4
6 x  4 
6  4  24

or
24 ln e  x
241  x
24  x
Solve – Round to the nearest hundredth. (2 Ways)
1.7  7
x
log 1.7  log 7
x
x log 1.7  log 7
log 7
x
log 1.7
x  3.67
ln 1.7  ln 7
x ln 1.7  ln 7
ln 7
x
ln 1.7
x  3.67
x
or
Solve – Round to the nearest hundredth. (Use ln)
e 8
x
ln e  ln 8
x ln e  ln 8
x1  ln 8
x  ln 8
x  2.08
x
Solve – Round to the nearest hundredth.
2e  18
x
x
2e
18

2
2
x
e 9
x ln e  ln 9
x  ln 9
x  2.20
Solve – Round to the nearest hundredth.
12e  48
3x
e 4
3 x ln e  ln 4
3 x  ln 4
ln 4
x
3
x  0.46
3x
Solve – Round to the nearest hundredth.
4.5e
x 7
 90
x 7
e  20
x  7 ln e  ln 20
x  7  ln 20
x  ln 20  7
x  4.00