20 ## LOGARITHMSWHEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 2 2 Inversely, if we are given the base 2 and its power 8 -- 2 -- then what is the exponent that will produce 8? That exponent is called a logarithm. We call the exponent 3 the logarithm of 8 with base 2. We write 3 = log We write the base 2 as a subscript. 3 is the A logarithm is an exponent. Since 10 then log "The logarithm of 10,000 with base 10 is 4." 4 is the "10 "log Here is the definition: log That base with that exponent produces
Example 1. Write in exponential form: log
Problem 1. Which numbers have negative logarithms? To see the answer, pass your mouse over the colored area. Proper fractions.
Example 3. Evaluate log log We can observe that, in any base, the logarithm of 1 is 0. log
Example 4. Evaluate log log In any base, the logarithm of the base itself is 1. log
Example 5 . log log The following is an important formal rule, valid for any base log This rule embodies the very meaning of a logarithm. The rule also shows that the exponential function
Compare the previous rule.
Example 7. log log Problem 2. Write each of the following in logarithmic form. To see the answer, pass your mouse over the colored area.
Problem 3. Write each of the following in exponential form.
Problem 4. Evaluate the following.
Problem 5. What number is
Problem 6. log Problem 7. Evaluate the following.
Common logarithms The system of common logarithms has 10 as its base. When the base is not indicated, log 100 = 2 then the system of common logarithms -- base 10 -- is implied. Here are the powers of 10 and their logarithms:
Logarithms replace a geometric series with an arithmetic series.
Problem 8. log 10
Problem 9. log 58 = 1.7634. Therefore, 10 58. 1.7634 is the common logarithm of 58. When 10 is raised to that exponent, 58 is produced.
Problem 10. log (log
log
The three laws of logarithms
"
"
_{b}x" We will prove these laws below.
according to the third law. The Answer above shows the complete theoretical steps. In practice, however, it is not necessary to write the line
The student should be able to go immediately to the next line --
-- if not to the very last line
Example 9. Solve this equation for
By this technique, we can solve equations in which the unknown appears in the exponent. Problem 11. Use the laws of logarithms to rewrite the following.
Change of base Say that we know the values of logarithms of base 10, but not, for example, in base 2. Then we can convert a logarithm in base 10 to one in base 2 -- or any other base -- by realizing that the values will be proportional. Each value in base 2 will differ from the value in base 10 by the same constant Now, to find that constant, we know that Therefore, on putting That implies Therefore, That is, By knowing the values of logarithms in base 10, we can in this way calculate their values in base 2. In general, then, if we know the values in base Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |