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38 LOGARITHMS A LOGARITHM is an exponent. Since 23 = 8, then 3 is called the logarithm of 8 with base 2. We write 3 = log28. 3 is the exponent to which 2 must be raised to produce 8. We write the base 2 as a subscript. Thus a logarithm is the exponent to which the base must be raised to produce a given number. Since 104 = 10,000, then log1010,000 = 4. "The logarithm of 10,000 with base 10 is 4." 4 is the exponent to which the base 10 must be raised to produce 10,000. "104 = 10,000" is called the exponential form. "log1010,000 = 4" is called the logarithmic form. Here is the definition: logbx = n means bn = x. That base with that exponent produces x. Example 1. Write in exponential form: log232 = 5 Answer. 25 = 32
Problem 1. Which numbers have negative logarithms? To see the answer, pass your mouse over the colored area. Proper fractions. Example 3. Evaluate log81. Answer. 8 to what exponent produces 1? 80 = 1. log81 = 0. We can observe that in any base b, the logarithm of 1 is 0. logb1 = 0 Example 4. Evaluate log55. Answer. 5 to what exponent will produce 5? 51 = 5. log55 = 1. In any base, the logarithm of the base itself is 1. logbb = 1 Example 5. log22m = ? Answer. 2 raised to what exponent will produce 2m ? m, obviously. log22m = m. This is an important formal rule, valid for any base b: logbbx = x This rule embodies the very meaning of a logarithm. x -- on the right -- is the exponent to which the base b must be raised.
Example 7. log2 .25 = ? Answer. .25 = ¼ = 2−2. Therefore, log2 .25 = log22−2 = −2 Example 8. log3 = ? Answer.. = 3. (Definition of a fractional exponent.) Therefore, log3 = log33 = 1/5 Problem 2. Write each of the following in logarithmic form.
Problem 3. Write each of the following in exponential form.
Problem 4. Evaluate the following.
Problem 5. What number is n?
Problem 6. logbbx = x Problem 7. Evaluate the following.
The three laws of logarithms 1. logbxy = logbx + logby "The logarithm of a product is equal to the sum
"The logarithm of a quotient is equal to the logarithm of the numerator 3. logb x n = n logbx "The logarithm of a power of x is equal to the exponent of that power For a proof of these laws, see Topic 20 of Precalculus.
Answer. According to the first two laws,
Now, = y½. Therefore, according to the third law,
Example 2. Use the laws of logarithms to rewrite log (sin x log x) Solution. This has the form log ab. a = sin x, b = log x. Therefore, log (sin x log x) = log sin x+ log log x Example 3. Use the laws of logarithms to rewrite log . Solution.
Problem 8. Use the laws of logarithms to rewrite 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. a) log 105 = 5. 10 is the base. b) log 10n = n c) log 58 = 1.7634. Therefore, 101.7634 = 58 1.7634 is the common logarithm of 58. When 10 is raised to that exponent, 58 is produced. Problem 9. log (log x) = 1. What number is x? log a = 1, implies a = 10. (See above.) Therefore, log (log x) = 1 implies log x = 10. Since 10 is the base, x = 1010 = 10,000,000,000 Example 4. Given: log 3 = .4771 Evaluate a) log 3000 Solution. Write 3000 in scientific notation:
b) log .003
Problem 10 Given: log 6 = .7781 Use the laws of logarithms to evaluate the following.
Example 5. Given: log 2 = .3010, log 3 = .4771 Evaluate log 18. Solution. 18 = 2· 3². Therefore,
Problem 11. Given: log 2 = .3010 log 3 = .4771 log 5 = .6990 Use the laws of logarithms to find the following. a) log 6 = log 2 + log 3 = .7781 b) log 15 = log 3 + log 5 = 1.1761 c) log 4 = log 2² = 2 log 2 = .6020 d) log 8 = log 2³ = 3 log 2 = .9030 e) log 30 = log 3 + log 10 = 1.4771 f) log 300 = log 3 + log 100 = 2.4771 g) log 3000 = log 3 + log 1000 = 3.4771 h) log 12 = log 3 + log 4 = 1.0791
j) log = ½ log 3 = .2386 k) log = ½ log 5 = .3495
n) log = ½(log 2 − log 3) = −.0881 o) log 1500 = log 3 + log 5 + log 100 = 3.1761 For the system of natural logarithms, see Topic 20 in Precalculus. Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |