Chapter 5: Exponential and Logarithmic Functions 5.5: Properties and Laws of Logarithms Essential Question: What are the three properties that simplify logarithmic expressions? Describe how to use them.

5.5: Properties and Laws of Logarithms Basic Properties of Logarithms ◦ Logarithms are only defined for positive real numbers  Not possible for 10 or e to be taken to an exponent and result in a negative number ◦ Log 1 = 0 and ln 1 = 0  10 0 = 1 & e 0 = 1 ◦ Log 10 k = k and ln e k = k  log = k  10 k = 10 4  k = 4 ◦ 10 log v = v and e ln v = v  10 log 22 = v  log 10 v = log 22  v = 22

5.5: Properties and Laws of Logarithms Solving Equations by Using Properties of Logarithms ◦ ln(x + 1) = 2  Method #1  e 2 = x + 1  e 2 – 1 = x  x ≈  Method #2  e ln(x + 1) = e 2  x + 1 = e 2  See method #1 above

5.5: Properties and Laws of Logarithms Product Law of Logarithms ◦ Law of exponents states b m b n = b m+n ◦ Because logarithms are exponents:  log (vw) = log v + log w  ln (vw) = ln v + ln w  Proof:  vw = 10 log v 10 log w = 10 log v + log w  vw = 10 log vw  Taking from above:  10 log v + log w = 10 log vw  log v + log w = log vw  Proof of ln/e works the same way

5.5: Properties and Laws of Logarithms Product Law of Logarithms (Application) ◦ Given that log 3 = and log 11 = find log 33  log 33= log (3 11) = log 3 + log 11 = = ◦ Given that ln 7 = and ln 9 = find ln 63  ln 63= ln (7 9) = ln 7 + ln 9 = =

5.5: Properties and Laws of Logarithms Quotient Law of Logarithms ◦ Law of exponents states ◦ Because logarithms are exponents:  log ( ) = log v – log w  ln ( ) = ln v – ln w ◦ Proof is the same as the Product Law

5.5: Properties and Laws of Logarithms Quotient Law of Logarithms (Application) ◦ Given that log 28 = and log 7 = find log 4  log 4= log (28 / 7) = log 28 – log 7 = – = ◦ Given that ln 18 = and ln 6 = find ln 3  ln 3= ln (18 / 6) = ln 18 – ln 6 = – =

5.5: Properties and Laws of Logarithms Power Law of Logarithms ◦ Law of exponents states (b m ) k = b mk ◦ Because logarithms are exponents:  log (v k ) = k log v  ln (v k ) = k ln v  Proof:  v = 10 log v → v k = (10 log v ) k = 10 k log v  v k = 10 log v k  Taking from above:  10 k log v = 10 log v k  k log v = log v k  Proof of ln/e works the same way

5.5: Properties and Laws of Logarithms Power Law of Logarithms (Application) ◦ Given that log 6 = find log  log = log 6 ½ = ½ log 6 = ½ (0.7782) = ◦ Given that ln 50 = find ln 

5.5: Properties and Laws of Logarithms Simplifying Expressions ◦ Write as a single logarithm:  ln 3x + 4 ln x – ln 3xy

5.5: Properties and Laws of Logarithms Simplifying Expressions ◦ Write as a single logarithm: 

5.5: Properties and Laws of Logarithms Assignment ◦ Page 369 ◦ Problems 1-25, odd problems ◦ Show work