1 / 12

5.1

5.1. LOGARITHMS AND THEIR PROPERTIES. What is a Logarithm?. If x is a positive number, log x is the exponent of 10 that gives x . In other words, if y = log x then 10 y = x. log 100 = 2 because 10 2 = 100.

caraf
Download Presentation

5.1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 5.1 LOGARITHMS AND THEIR PROPERTIES

  2. What is a Logarithm? If x is a positive number, log x is the exponent of 10 that gives x. In other words, if y = log x then 10y = x. log 100 = 2 because 102 = 100 log 2500 ≈ 3.398 because 103.398= 2500

  3. Logarithms to Exponents Example 1 Rewrite the following statements using exponents instead of logs. • log 100 = 2 (b) log 0.01 = −2 (c) log 30 = 1.477 Solution We use the fact that if y = log x then 10y = x. (a) 2 = log 100 means that 102 = 100. (b) −2 = log 0.01 means that 10−2 = 0.01. (c) 1.477 = log 30 means that 101.477 ≈ 30.

  4. Exponents to Logarithms Example 2 Rewrite the following statements using logs instead of exponents. (a) 105 = 100,000 (b) 10−4 = 0.0001 (c) 10.8 ≈ 6.3096. Solution We use the fact that if 10y = x, then y = log x. (a) 105 = 100,000 means that log 100,000 = 5. (b) 10−4 = 0.0001 means that log 0.0001 = −4. (c) 10.8 = 6.3096 means that log 6.3096 ≈ 0.8.

  5. Logarithms Are Exponents Example 3 Without a calculator, evaluate the following, if possible: (a) log 1 (c) log 1,000,000 (d) log 0.001 (f) log(−100) Solution (a) We have log 1 = 0, since 100 = 1. (c) Since 1,000,000 = 106, the exponent of 10 that gives 1,000,000 is 6. Thus, log 1,000,000 = 6. (d) Since 0.001 = 10−3, the exponent of 10 that gives 0.001 is −3. Thus, log 0.001 = −3. (f) Since 10 to any power is positive, −100 cannot be written as a power of 10. Thus, log(−100) is undefined.

  6. Logarithmic and Exponential Functions are Inverses For any N, log(10N) = N and for N > 0, 10logN = N.

  7. Properties of Logarithms Properties of the Common Logarithm • By definition, y = log x means 10y = x. • In particular, log 1 = 0 and log 10 = 1. • The functions 10x and log x are inverses, so they “undo” each other: log(10x) = x for all x, 10log x= x for x > 0. • For a and b both positive and any value of t, log(a b) = log a + log b log(a/b) = log a − log b log(bt) = t · log b.

  8. Applying Properties of Logarithms Example 5 Solve 100・2t= 337,000,000 for t. Solution Dividing both sides of the equation by 100 gives 2t = 3,370,000. Taking logs of both sides gives log 2t= log(3,370,000). Since log(2t) = t · log 2, we have t log 2 = log(3,370,000), so, solving for t, we have t = log(3,370,000)/log 2 = 21.684.

  9. The Natural Logarithm For x > 0, lnx is the power of e that gives x or, in symbols, lnx = y means ey= x, and y is called the natural logarithm of x.

  10. Same Properties as for Common Logarithm Properties of the Natural Logarithm • By definition, y = lnx means ey = x. • In particular, ln 1 = 0 and lne = 1. • The functions ex and ln x are inverses, so they “undo” each other: ln(ex) = x for all x, elnx= x for x > 0. • For a and b both positive and any value of t, ln (a b) = lna + lnb ln (a/b) = lna − lnb ln (bt) = t · lnb.

  11. Applying Properties of Natural Logarithms Example 6 (a) Solve for x: 5e2x= 50 Solution (a) We first divide both sides by 5 to obtain e2x = 10. Taking the natural log of both sides, we have

  12. Misconceptions and Calculator Errors Involving Logs NO NONO • log(a + b) is not the same as log a + log b • log(a − b) is not the same as log a − log b • log(a b) is not the same as (log a)(log b) • log (a/b) is not the same as (log a) / (log b) • log (1/a) is not the same as 1 / (log a).

More Related