For the problem introducing the previous section, we found that the time required is weeks. Some calculators will give this directly as about 3.32 weeks (ie 3 weeks 2 days and about 6 hours), but even with such calculators it may take longer to figure out how to enter it properly than to simplify the expression before evaluating. In this section you will learn how to combine and simplify expressions involving logarithms and in fact will find that just a single log button is all you need.
After completing this section you should be able to:
1. Review (or just recall to mind) the laws of exponents. For each of these we’ll get a translation into the language of logarithms.
2. Use the Reading Guide to section 4.5 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.
3. Read the following Study Notes and Discussion, and make sure that you have absorbed the main points by answering the questions.
4. Follow
the instructions regarding Further Practice and when ready
move on to the next section.
Reading Guide
p330-331
Each of the laws of logarithms is just the translation into logarithm language of one of the laws of exponents:
gives
so is the power of a needed to give result uw which is just
gives
so is the power of a needed to give result u/w which is just
gives
so is the power of a needed to give result which is just
Historically (before calculators) logarithms (usually common logs) were listed in books of tables so that people could reduce the amount of work needed to multiply multidigit numbers by looking up their logs, adding the logs (which takes fewer steps than multiplying by hand) and looking up the sum in a table of “antilogs” (just the same table listed the other way around). But these laws are also useful for simplifying algebraic expressions and solving equations as we shall see.
p332
The laws of logarithms can be used either to express the logarithm of a single complicated expression in terms of logarithms of its component factors (as in Example#1 on page 332), or to go the other way (as in Example#2). The latter (i.e. combining logs) is often useful for solving equations involving logarithms.
p333-4
Examples#3&4&5 all involve combining logs (with respect to the same base) into a single log so that the equation either becomes giving A=B or giving .
p334
In Example#6 note that (although it’s not easy to see because of the scale of the picture) the solution could also be obtained by squeezing the graph of sideways to 1/81 of its width. For a better illustration of this connection between shifting and scaling you might look at the graph of which comes from by either shifting down by one or stretching sideways by a factor of three.
p335
In Example #7, we see that the graphs of and agree for x>0. Another equation that gives the same graph for x>0 is . This is because two factors of are needed to match one of 3, so the exponent on needed to produce a result is twice that needed on 3, so . This illustrates another version of the power of power rule, namely (with b=3 and ).
This differs from the other laws of logs in that it relates logs with different bases, and in the next section we’ll see how the same idea allows us to relate any two bases and so to express all logs in terms of logs with respect to any fixed base.
The main points to remember are as follows:
Each of the laws of logarithms is just the translation into logarithm language of one of the laws of exponents:
gives
so is the power of a needed to give result uw which is just
gives
so is the power of a needed to give result u/w which is just
gives
so is the power of a needed to give result which is just
Historically (before calculators) logarithms (usually common logs) were listed in books of tables so that people could reduce the amount of work needed to multiply multidigit numbers by looking up their logs, adding the logs (which takes fewer steps than multiplying by hand) and looking up the sum in a table of “antilogs” (just the same table listed the other way around). But these laws are also useful for simplifying algebraic expressions and solving equations as we shall see.
The laws of logarithms can be used either to express the logarithm of a single complicated expression in terms of logarithms of its component factors (as in Example#1 on page 332), or to go the other way (as in Example#2). The latter (i.e. combining logs) is often useful for solving equations involving logarithms.
For the equation from the radiation problem in the Introduction to Sec 4.4, we saw that the solution is given by (by the definition of what a logarithm is), but we couldn’t see how to compute it exactly without a calculator that does general logarithms.
But using the ideas of this section, there is another way.
Taking logs of both sides of gives , and using the laws of logarithms we can rewrite this as . So we get
So we get the same answer as before, but this time using a calculator which has only common logs.
Laws of Logarithms
Each of the laws of logarithms is just the translation into logarithm language of one of the laws of exponents:
gives
so is the power of a needed to give result uw which is just
gives
so is the power of a needed to give result u/w which is just
gives
so is the power of a needed to give result which is just
Note:
1. There is no need to remember the book’s numbering of these as “law 1”, “law 2” and “law 3”. It is better to refer to them by what they say, e.g. “the log of product equals sum of logs rule”, and so on. How would you express the other two in words?
(see answer #1 below)
2. If you want to save a few brain cells, you might notice that you don’t really need to remember all three. For example can you see how to prove the middle one as a consequence of the other two? (see answer #2 below)
Applications of Laws of Logarithms
Historically (before calculators) logarithms (usually common logs) were listed in books of tables so that people could reduce the amount of work needed to multiply multidigit numbers by looking up their logs, adding the logs (which takes fewer steps than multiplying by hand) and looking up the sum in a table of “antilogs” (just the same table listed the other way around). But these laws are also useful for simplifying algebraic expressions and solving equations, as we shall see.
The historic use of logarithms described above was based on using the laws of logarithms to express the logarithm of a single complicated expression in terms of logarithms of its component factors (as in Example#1 on page 332 of the text). The laws can also be used to go the other way (as in Example#2). The latter (i.e. combining logs) is often useful for solving equations involving logarithms.
As an example, try to solve the equation . (see answer #3 below)
For another example, consider the equation from the radiation problem introduced earlier in this unit. We saw that the solution is given by (by the definition of what a logarithm is), but we couldn’t see how to compute it exactly without a calculator that does general logarithms.
But using the ideas of this section, there is another way.
Taking logs of both sides of gives , and using the laws of logarithms we can rewrite this as . So we get
So we get the same answer as before, but this time using a calculator which has only common logs.
Check your understanding, and practice for speed, by working through some of the Exercises on pp336-338.
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, you should try ##1,21,31,41,51. Where you can’t match the solution in the guide without looking, check the solutions guide to see what you were missing and then try others of the same type. Continue until you are confident that you can do a similar set of five without assistance.
1. “log of quotient equals difference of logs”, and “log of power equals multiple of log”
2.
3. Using “sum of logs equals log of product” on the left, we get
So , which means . So , and .
For it is easy to check that the two sides of the original equation agree, but for the term on the left is undefined (negative argument in log).
So is the only solution.