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ELF.01.6 - Laws of Logarithms

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ELF.01.6 - Laws of Logarithms

MCB4U - Santowski

- if f(x) = ax, find f -1(x) so y = ax then x = ay 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 = ay as y = loga(x)

Consider this approach here in column 1

evaluate log5(625) = x

5x = 625

5x = 54

x = 4

Now try log5(625) = x tis way:

log5(54) = x

log5(5)4 = x

Now I know from column 1 that the answer is 4 … so …

log5(5)4 = 4

4 x log5(5) = 4

4 x 1 = 4

- So if log5(625) = log5(5)4 = 4 x log5(5)
- It would suggest a rule of logarithms
- logb(bx) = x
- Which we can generalize
- logb(ax) = xlogb(a)

- Consider the example
- Recall that the expression log3(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

- Let’s take our exponential equation and write it in logarithmic form
- So becomes log3(x) = log3(5)
- Since both sides of our equation have a log3 then x = 5 as we had tried to reason out in the previous slide
- So we can generalize that

- Express logam + logan as a single logarithm
- We will let logam = x and logan = y
- So logam + logan becomes x + y
- But if logam = x, then ax = m and likewise ay = n
- Now take the product (m)(n) = (ax)(ay) = ax+y
- Rewrite mn=ax+y in log form loga(mn)=x + y
- But x + y = logam + logan
- So thus loga(mn) = logam + logan

- We could develop a similar “proof” to the quotient law as we did with the product law except we ask you to express logam - logan as a single logarithm
- As it turns out (or as is predictable), the simplification becomes as follows:
- loga(m/n) = loga(m) – loga(n)

- (i) log354 + log3(3/2)
- (ii) log2144 - log29
- (iii) log30 + log(10/3)
- (iv) which has a greater value
- (a) log372 - log38 or (b) log500 + log2

- (v) express as a single value
- (a) 3log2x + 2log2y - 4log2a
- (b) log3(x+y) + log3(x-y) - (log3x + log3y)

- (vi) log2(4/3) - log2(24)
- (vii) (log25 + log225.6) - (log216 + log39)

- Logarithm Rules Lesson from Purple Math
- College Algebra Tutorial on Logarithmic Properties from West Texas AM

- Nelson textbook, p125, Q1-14