Logarithm

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Prepared for Gina. L

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1)  AB = C   ---->   B =logA(C)

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Proof

AB = C

log AB = log C  -- take log on both sides

B log A = log

B = log C / log A

B =log A(C)

 

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2)  logb(mn) = n P logb(m)

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example 1.1

5 log5(27)

= 5 log5(33)

= (5 3) log5(3)

= 15 log5(3)

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3)  logb(mn) = logb(m) + logb(n)

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example 2.1

log2(20) 

= log2(4 5) 

= log2(4) + log2(5) 

= log2(22) + log2(5)

= 2 log2(2) + log2(5)

= 2 + log2(5)

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*Notice: logA(A) = logA / logA = 1

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4)  logb(m/n) = logb(m) V logb(n)

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example 3.1

log2(15/4)

=  log2(15) V log2(4)

=  log2(15) V log2(22)

=  log2(15) V 2log2(2)

=  log2(15) V 2

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5)  loga(m) = 1 / logm(a

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example 4.1

log2(9)

= 1 / log9(2)

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example 4.2

log12(2) log2(144)

=  [1 / log2(12)] log2(144)

= log2(144) / log2(12)

= log12(144)

= log12(122)

= 2log12(12)

= 2

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6) For base = e, 2 and 10

loge(m) = ln (m)

log2(m) = lg (m)

log10(m) = log (m)

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note 5.1

Mathematical Constant e = 2.71828.... (irrational)

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7) Differentiation

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d/dx loga(x) = (1/x) (1/lna)

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example 6.1

d/dx log3(x)

= (1/x) (1/ln3)

= 1/(x ln3)

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d/dx loga(g(x)) = (1/g(x)) (1/lna) g'(x)

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example 6.2

d/dx log5(g(x)),  given g(x) = 2x2

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d/dx log5(g(x))

= d/dx log5(2x2)

= 1/(2x2) 1/ln5 d/dx (2x2)

= 1/(2x2) 1/ln5 4x

= (4x)/(2x2 ln5)

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d/dx ln(x) = 1/x

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example 6.3

d/dx ln3(4x5)

= d/dx (ln(4x5))3  

= 3 (ln(4x5))3-1 d/dx ln(4x5)

= 3 (ln(4x5))2 (1/(4x5)) d/dx (4x5)

= 3 (ln(4x5))2 (1/(4x5)) (20x4)

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P.S.

Notice the following two equations are NOT the same.

ln3(m) = ln (m) ln (m) ln (m)

ln(m3) = ln (m m m) = 3 ln(m)

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As well as these two equations.

Sin2(c) = Sin (c) Sin (c)

Sin(c2) = Sin (c c)

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written by : Kevin T (Mar 8, 2010)

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