Module 4 
Section 4.5
Introduction
In previous sections you have learned how to solve some exponential and logarithmic equations. This is not surprising as logarithms are defined as solutions to exponential equations! You have also acquired some skills in simplifying and rearranging expressions involving powers and logarithms. In this section you will extend these skills so as to be able to solve more complicated equations that may arise in some of the applications we’ve seen, and/or in other applications that you may come across in future.
Section Learning Outcomes
After completing this section you should be able to:
- Use the change of base law to express logs with respect to any base in terms of logs with respect to any other
- Solve a wide variety of equations involving exponentials and logarithms
What to Do
1. Follow the Reading Guide below for section 4.6 of the text. Keep a pencil and paper at hand, and be sure to check your understanding by trying the examples before reading how to do them.
2. Read the following Summary Notes and Discussion, and make sure that you have absorbed the main points by answering the questions.
3. Follow the instructions regarding Practice Exercises.
Reading
p338-340
In the last Section we saw that the
solution of our radiation problem could also be written
as
.
So
.
This is a special case of the general change of base formula discussed on pages 338-9.
The change of base formula is really just another way of writing the rule that taking a power of a power corresponds to multiplication of exponents.
p340-5
Some of the steps in Examples#4-7 on pages 341-345 should remind
you of how you solved equations like or
by reducing to quadratics back in Module 3.
p346-7
The ‘logistic function’ discussed in Examples#10&11 (on pages 346-7) is a good model for population growth in a finite environment where the limited supply of food causes some to starve and prevents the unlimited growth of the simple exponential model. Determining the time required to reach a given population as a function of that population size amounts to finding the inverse function, and can be done by similar steps to those used in Example#7.
Summary Notes and Discussion
Change of Base
In the last Section we saw that the
solution of our radiation problem could also be written
as
.
So
.
This is a special case of the general change of base formula discussed on pages 339-340 of the text.
What could you do if your calculator only had a ln button and no common log?
(see answer#1)
The change of base formula is really just another way of writing the rule that taking a power of a power corresponds to multiplication of exponents.
More Equations
In the examples in the text on solving
equations, you should have noticed that equations involving exponentials are
typically solved by taking logs at some point but not necessarily at the beginning. In
general we have nice rules for dealing with sums of logs or logs of products,
but not for logs of sums. So it is not a good idea to take logs until you have
got the equation in a form with a single term (which may itself be a product) on each side of the equation.
Some of the steps in Examples#4-7 of the text should have reminded
you of how you solved equations like or
by reducing to quadratics back in Module 3 (by
multiplying by x in the first case
and making the substitution
in the second).
The ‘logistic function’ discussed in Examples#10&11 (on pages 346-7) is a good model for population growth in a finite environment where the limited supply of food causes some to starve and prevents the unlimited growth of the simple exponential model. Determining the time required to reach a given population as a function of that population size amounts to finding the inverse function, and can be done by similar steps to those used in Example#7.
Try to find for
for yourself before looking below (at answer#2).
Further Practice
Check your understanding and practice for speed by working through some of the Exercises on pp348-351.
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort)
As a minimum, start with ##1,11,21,31,41,51.
Answers to Questions
1. Use
2. If
then
So ,
giving
,
and so
And now, taking ln of both sides, ,
so
But y
was just . So
.