Unit 3.5

3.5
Equation
Solving and
Modeling
Copyright © 2011 Pearson, Inc.

What you’ll learn about
 Solving Exponential Equations
 Solving Logarithmic Equations
 Orders of Magnitude and Logarithmic Models
 Newton’s Law of Cooling
 Logarithmic Re-expression
… and why
The Richter scale, pH, and Newton’s Law of Cooling, are
among the most important uses of logarithmic and
exponential functions.
Copyright © 2011 Pearson, Inc. Slide 3.5 - 2

One-to-One Properties
For any exponential function f (x)  bx ,
g If bu  bv , then u  v.
For any logarithmic function f (x)  logb x,
g If logb u  logb v, then u  v.

Example Solving an Exponential
Equation Algebraically
Solve 401/ 2x/2
 5.

Example Solving an Exponential
Equation Algebraically
Solve 401/ 2x/2
401 / 2x/2
 5.
 5
1 / 2x/2

1
8
divide by 40
1
2

 

 
x/2

1
2

 

 
3
1
8

1
2

 

 
3
x / 2  3 one-to-one property
x  6

Example Solving a Logarithmic
Equation
Solve log x3  3.

Equation
Solve log x3  3.
log x3  3
log x3  log103
x3  103
x  10

Equation
Solve log2x 1 lnx  3 ln8  2x

Equation
Solve Graphically
To use the x-intercept method, we rewrite the equation
log2x 1 lnx  3 ln8  2x 0
and graph f x log2x 1 lnx  3 ln8  2x.
The x-intercept is x 
1
2
,
which is a solution to the equation.

Equation
Confirm Algebraically
log2x 1 lnx  3 ln8  2x
log 2x 1   x  3     ln 8  2x  
2x 1x  3 8  2x
2x2  9x  5  0
2x 1x  5 0 x 
1
2
or x  5
Notice x  5 is an extraneous solution. So the only solution
is x 
1
2
.

Orders of Magnitude
The common logarithm of a positive quantity is its order
of magnitude.
Orders of magnitude can be used to compare any like
quantities:
 A kilometer is 3 orders of magnitude longer than a
meter.
 A dollar is 2 orders of magnitude greater than a penny.
 New York City with 8 million people is 6 orders of
magnitude bigger than Earmuff Junction with a
population of 8.

Richter Scale
The Richter scale magnitude R of an earthquake is
R  log
a
T
 B, where a is the amplitude in
micrometers (m) of the vertical ground motion at
the receiving station, T is the period of the associated
seismic wave in seconds, and B accounts for the
weakening of the seismic wave with increasing
distance from the epicenter of the earthquake.

pH
In chemistry, the acidity of a water-based solution is
measured by the concentration of hydrogen ions in the
solution (in moles per liter). The hydrogen-ion
concentration is written [H+]. The measure of acidity
used is pH, the opposite of the common log of the
hydrogen-ion concentration:
pH = –log [H+]
More acidic solutions have higher hydrogen-ion
concentrations and lower pH values.

Newton’s Law of Cooling
An object that has been heated will cool to the temperature
of the medium in which it is placed. The temperature T of
the object at time t can be modeled by
T (t )  Tm  (T0  Tm )e kt
for an appropriate value of k, where
Tm  the temperature of the surrounding medium,
T0  the temperature of the object.
This model assumes that the surrounding medium maintains
a constant temperature.

Example Newton’s Law of Cooling
A hard-boiled egg at temperature 100ºC is placed in
15ºC water to cool. Five minutes later the temperature
of the egg is 55ºC. When will the egg be 25ºC?

Example Newton’s Law of Cooling
A hard-boiled egg at temperature 100ºC is placed in
15ºC water to cool. Five minutes later the temperature
of the egg is 55ºC. When will the egg be 25ºC?
Given T0  100, Tm  15, and T (5)  55.
T (t )  Tm  (T0  Tm )e kt
55  15  85e5k
40  85e5k
40
85

 

 
 e5k
ln
40
85



   5k
k  0.1507...
Now find t when T (t )  25.
25  15  85e0.1507t
10  85e0.1507t
ln
10
85

 

 
 0.1507t
t  14.2min.

Regression Models Related by
Logarithmic Re-Expression
 Linear regression: y = ax + b
 Natural logarithmic regression: y = a + blnx
 Exponential regression: y = a·bx
 Power regression: y = a·xb

Three Types of Logarithmic
Re-Expression

Re-Expression (cont’d)

Quick Review
Prove that each function in the given pair is the inverse
of the other.
1. f (x)  e3x and g(x)  ln x1/3  
2. f (x)  log x2 and g(x)  10x/2
Write the number in scientific notation.
3. 123,400,000
Write the number in decimal form.
4. 5.67 108
5. 8.9110-4

Quick Review Solutions
Prove that each function in the given pair is the inverse
of the other.
1. f (x)  e3x and g(x)  ln x1/3   f (g(x))  e
3ln x1/3  
 elnx  x
2. f (x)  log x2 and g(x)  10x/2 f (g(x))  log 10x/2  2
 log10x  x
Write the number in scientific notation.
3. 123,400,000 1.234 108
Write the number in decimal form.
4. 5.67 108 567,000,000
5. 8.9110-4 0.000891

Unit 3.5

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Unit 3.5