1. ELF.01.6 - Laws of Logarithms MCB4U - Santowski

2. (A) Review ► if f(x) = a x, find f -1 (x) so y = a x then x = a y and now isolate y ► in order to isolate the y term, the logarithmic notation or symbol or function was invented or created so we write x = a y as y = log a (x)

3. (B) Properties of Logarithms- Logarithms of Powers ► Consider this approach here in column 1 ► evaluate log 5 (625) = x ► 5 x = 625 ► 5 x = 5 4 ► x = 4 ► Now try log 5 (625) = x tis way: ► log5(5 4 ) = x ► log 5 (5) 4 = x ► Now I know from column 1 that the answer is 4 … so … ► log 5 (5) 4 = 4 ► 4 x log 5 (5) = 4 ► 4 x 1 = 4

4. (B) Properties of Logarithms- Logarithms of Powers ► So if log 5 (625) = log 5 (5) 4 = 4 x log 5 (5) ► It would suggest a rule of logarithms  ► log b (b x ) = x ► Which we can generalize  ► log b (a x ) = xlog b (a)

5. (C) Properties of Logarithms – Logs as Exponents ► Consider the example ► Recall that the expression log 3 (5) simply means “the exponent on 3 that gives 5”  let’s call that x ► So we are then asking you to place that same exponent (the x) on the same base of 3 ► Therefore taking the exponent that gave us 5 on the base of 3 (x ) onto a 3 again, must give us the same 5!!!! ► We can demonstrate this algebraically as well

6. (C) Properties of Logarithms – Logs as Exponents ► Let’s take our exponential equation and write it in logarithmic form ► So becomes log 3 (x) = log 3 (5) ► Since both sides of our equation have a log 3 then x = 5 as we had tried to reason out in the previous slide ► So we can generalize that

7. (D) Properties of Logarithms – Product Law ► Express log a m + log a n as a single logarithm ► We will let log a m = x and log a n = y ► So log a m + log a n becomes x + y ► But if log a m = x, then a x = m and likewise a y = n ► Now take the product (m)(n) = (a x )(a y ) = a x+y ► Rewrite mn=a x+y in log form  log a (mn)=x + y ► But x + y = log a m + log a n ► So thus log a (mn) = log a m + log a n

8. (E) Properties of Logarithms – Quotient Law ► We could develop a similar “proof” to the quotient law as we did with the product law  except we ask you to express log a m - log a n as a single logarithm ► As it turns out (or as is predictable), the simplification becomes as follows: ► log a (m/n) = log a (m) – log a (n)

9. (F) Summary of Laws Logs as exponents Product Rule log a (mn) = log a m + log a n Quotient Rule log a (m/n) = log a (m) – log a (n) Power Rule Log a (m p ) = (p) x (log a m)

10. (G) Examples ► (i) log 3 54 + log 3 (3/2) ► (ii) log 2 144 - log 2 9 ► (iii) log30 + log(10/3) ► (iv) which has a greater value  (a) log 3 72 - log 3 8 or (b) log500 + log2 ► (v) express as a single value  (a) 3log 2 x + 2log 2 y - 4log 2 a  (b) log 3 (x+y) + log 3 (x-y) - (log 3 x + log 3 y) ► (vi) log 2 (4/3) - log 2 (24) ► (vii) (log 2 5 + log 2 25.6) - (log 2 16 + log 3 9)

11. (G) Internet Links ► Logarithm Rules Lesson from Purple Math Logarithm Rules Lesson from Purple Math Logarithm Rules Lesson from Purple Math ► College Algebra Tutorial on Logarithmic Properties from West Texas AM College Algebra Tutorial on Logarithmic Properties from West Texas AM College Algebra Tutorial on Logarithmic Properties from West Texas AM

12. (H) Homework ► Nelson textbook, p125, Q1-14