# ELF.01.6 - Laws of Logarithms - PowerPoint PPT Presentation

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ELF.01.6 - Laws of Logarithms. MCB4U - Santowski. (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).

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ELF.01.6 - Laws of Logarithms

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## ELF.01.6 - Laws of Logarithms

MCB4U - Santowski

### (A) Review

• 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

### (B) Properties of Logarithms- Logarithms of Powers

• 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)

### (C) Properties of Logarithms – Logs as Exponents

• 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

### (C) Properties of Logarithms – Logs as Exponents

• 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

### (D) Properties of Logarithms – Product Law

• 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

### (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 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)

### (G) Examples

• (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)

### (G) Internet Links

• Logarithm Rules Lesson from Purple Math

• College Algebra Tutorial on Logarithmic Properties from West Texas AM

### (H) Homework

• Nelson textbook, p125, Q1-14