J.7 Properties of Logarithms

Transcription

1 J.7. PROPERTIES OF LOGARITHMS 1 J.7 Properties of Logarithms J.7.1 Understanding Properties of Logarithms Product Rule of Logarithms log a MN = log a M +log a N Example J.7.1. Rewrite as a sum of logarithms: log Solution. Since 3 is the base and 6 and 5 are the factors, we see in the formula log a MN, a = 3, M = 6, and N = 5. Hence, log = log 3 6+log 3 5 Recall. The logarithm with base e, where e is Euler s constant, is called the natural logarithm, i.e., log e x = lnx = y The number e on your calculator gives the value e We treat natural logarithms the same way, but in this case, the base a = e. Example J.7.. Rewrite as a sum of logarithms: lnk Solution. Since e is the base and and k are the factors you see this when we write k as k, we see in the formula log a MN, a = e, M =, and N = k. Hence, lnk = log e k = log e +log e k = ln+lnk Quotient Rule of Logarithms M log a = log N a M log a N Example J.7.3. Rewrite as a difference of logarithms: log Solution. Since 3 is the base, 7 is the numerator, and 5 is the denominator, we see in the M formula log a, a = 3, M = 7, and N = 5. Hence, N 7 log 3 = log log 3 5 Becarefultoobservethatthevalue of the log after the minussignisthevalue of the denominator of the fraction.

2 Example J.7.4. Rewrite as a difference of logarithms: ln 7 Solution. Since e is the base, 7 is the numerator, and is the denominator, we see in the M formula log a, a = e, M = 7, and N =. Hence, N ln 7 = log e 7 = log e7 log e = ln7 ln Be careful to observe that the value of the log after the minus sign is the value of the denominator of the fraction. Power Rule of Logarithms log a M p = plog a M Example J.7.5. Rewrite all powers as factors: log 7 4. Solution. Since 4 is the power on, then we can bring down 4 in front of the log: log 7 4 = 4 log 7 = 4log 7 = notice 4 and log 7 become factors. Example J.7.6. Rewrite all powers as factors: lnx. Solution. Since is the power on x, then we can bring down in front of the ln: lnx = lnx = lnx = notice and lnx become factors. J.7. Expanding and Contracting Logarithms Rule of Thumb for Expanding Logarithms When expanding logarithms, be sure to Rule 1. write all products as sums Rule. write all quotients as differences Rule 3. write all powers as factors Example J.7.7. Expand the logarithm by rewriting as a sum or difference of logarithms with 1000 x powers as factors. log. y Solution. We seeaquotient forthevalueofthelogarithm, soweforeseewewill usethequotient Rule of Logarithms. If we look closer at the numerator, we see there is a product of two factors.

3 J.7. PROPERTIES OF LOGARITHMS 3 Hence, we will use the Product Rule of Logarithms, too. I m not sure how good your eyes are, but I also foresee we will have to use the Power Rule of Logarithms x log = log 1000 x logy = Quotient Rule of Logarithms y = log1000+log x logy = Product Rule of Logarithms = log1000+logx 1/ logy = write x as x 1/ = log logx logy = Power Rule of Logarithms Notice, we had to rewrite x as x 1/ in order to see there was a power on x in which we had to use the Product Rule of Logarithms to bring it down. Thus, using the Rule of Thumb, all products are written as sums, all quotients are written as differences, and all powers are written as factors. Rule of Thumb for Contracting Logarithms When contracting logarithms, be sure to Rule 1. write all factors as powers Rule. write all sums as products Rule 3. write all differences as quotients Example J.7.8. Write log 9+log x log x 4 as a single logarithm. Solution. Right away we see a sum and difference with logarithms, so we know we will use Quotient and Product Rule of Logarithms. I m not sure how good your eyes are, but I also foresee we will have to use the Power Rule of Logarithms...can you notice when? log 9+log x log x 4 = log 9+log x log x 4 = Power Rule of Logarithms = log 9x log x 4 = Product Rule of Logarithms = log 9x x 4 = Quotient Rule of Logarithms Notice, we had to rewrite log x as log x in order to see there was a power on x in which we had to use the Product Rule of Logarithms to bring the up into the exponent. Thus, using the Rule of Thumb, all factors are written as powers usually, this is the first step, all sums are written as products, all differences are written as quotients. J.7.3 Change of Base Formula If a 1, b 1, and M are positive real numbers, then log a M = logm loga or log am = lnm lna Example J.7.9. Approximate log 9. Round your answer to three decimal places. Solution. We would like to approximate this value using a calculator, but we cannot easily enter a logarithm in base. We must rewrite log 9 so that we can easily enter it into the calculator.

4 4 This is where the Change of Base COB formula comes in handy. Notice the base a = and the value M = 9. Using the COB formula, we rewrite log 9 as log 9 = log9 log Recall, log is the common logarithm, log 10. Putting log9 log into the calculator, we get Note. We could have easily used the natural logarithm for the COB formula and would have obtained the same results. J.7.4 Other Properties of Logarithms log a 1 = 0 log a a = 1 a log a M = M log a a r = r Example J Evaluate each logarithm. a log 5 1 Solution. Since we need to find 5? = 1, then by the first property we know the result is zero. Thus, log 5 1 = 0. b log10 Solution. First, the log has no visible base. By default we use the common logarithm and assume the base is 10. So, since we need to find 10? = 10, then by the second property we know the result is one. Thus, log10 = 1. c log10 4 Solution. First, the log has no visible base. By default we use the common logarithm and assume the base is 10. So, since we need to find 10? = 10 4, then by the last property we know the result is 4. Thus, log10 4 = 4. d 1 log 1 1 Solution. If we rewrite this in logarithmic form, we get log 1? = log 1 1 We can easily see if this statement has to be true, then? = 1. Also, by the third property, we know the result is 1. Thus, 1 log 1 1 = 1.

5 J.7. PROPERTIES OF LOGARITHMS 5 Practice Write the expression as a logarithm of a single expression. Assume that variables represent positive numbers. 1. log a m log a n+6log a k. log 8 6+log 8 x 3. log 8 3+log 8 x 3 +log log a x+1 log a x 1+ Write as the sum and/or difference of logarithms. Express powers as factors log 4 x 1 x 6. log y 6 7. log b xz 3 xy 5 8. log b z 7 Use the Change of Base Formula and a calculator to evaluate the logarithm. Round to four decimal places. 9. log log log Evaluate each logarithm. 1. log log log log 1 1 3

6 6 Answers mk 6 1. log a n. log 8 6x 3. log 8 6x log a a x+1 3 x log 4x 1 6. log x 6log y 7. log b x+3log b z 8. log b x+5log b y 7log b z