Logarithm 25 June 08

7/31/2019 Logarithm 25 June 08

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Natural Logarithms

The system of logarithms whose base is the constant e is called the natural logarithm systemwhere ln 10 = 2.3026or that e = 2.3026

Logarithmsy = logb(x) if and only if x = by

logb(1) = 0logb(b) = 1log b(x*y) = logb(x) + logb(y)logb(x/y) = logb(x) logb(y)logb(x n) = n logb(x)logb(x) = logb(c) * logc(x) = logc(x) / logc(b)

http://www.math.com/tables/algebra/exponents.htm

Laws of Logarithm :1. logA + logB = logAB2. logAn= n logA3. logA logB = logA/B

Rules.log(xy) = log(x) + log(y) ln(xy) = ln(x) + ln(y)log(x/y) = log(x) - log(y) ln(x/y) = ln(x) - ln(y)log(xy) = ylog(x) ln(xy) = yln(x)10x10y = 10x+y exey = ex+y

10x/10y = 10x-y ex/ey = ex-y

(10x)y = 10xy (ex)y = exylog(1) = log(100) = 0 ln(1) = ln(e0) = 010logx = x = log(10x) elnx = x = ln(ex)

Using the logarithm and exponential functions

The key to using these functions is to remember how good they are at undoing each other: 10 logx = x = log(10x)elnx = x = ln(ex)

which is to say, applying first one and then the other (in either order) to a number returns the original number!

http://www.math.unh.edu/mac/calc/power.html

Product Rule for Logarithms

ForM> 0 andN> 0,log a (MN) = log a (M) + log a (N).Quotient Rule for LogarithmsForM> 0 andN> 0,

.log a (N) = log a(M)log a(N)

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http://www.math.com/tables/algebra/exponents.htmhttp://www.math.unh.edu/mac/calc/power.htmlhttp://www.math.com/tables/algebra/exponents.htmhttp://www.math.unh.edu/mac/calc/power.html


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MPower Rule for LogarithmsForM> 0 and any real numberN,

log a (MN) =N log a (M).

Inverse RulesIfa > 0 and a 1, then1. log a (a x) =x for any real numberx2. aloga (x)=x forx> 0.Rules of LogarithmsIfM, N, and a are positive real numbers with a 1,andx is any real number, then1. log a (a) = 1 2. log a (1) = 03. log a (a x) =x 4.

5. loga

(MN) = loga

(M) + loga

(N)6. log a (M/N) = log a (M)log a (N)7. log a (Mx ) =x log a (M)8. log a (1/N) =log a (N)Base-Change FormulaIfa > 0, b > 0, a 1, b 1, andM> 0, then

log a(M) = log b(M)/ log b (a)http://www.kemt.fei.tuke.sk/Predmety/KEMT320_EA/_web/Online_Course_on_Acoustics/lo garithms.html

Expansions of the Logarithm Function

Function Summation Expansion Comments

ln (x)=

(x-1)nn

= (x-1) - (1/2)(x-1)2 + (1/3)(x-1)3 + (1/4)(x-1)4 + ...

Taylor SeriesCentered at 1(0 < x 1/2)

ln (x)

=ln(a)+

(-1)n-1(x-a)n

n an

Taylor Series(0 < x


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= ln(a) + (x-a)/a - (x-a)2 / 2a2 + (x-a)3/3a3 - (x-a)4/4a4 +...

ln (x) =2

((x-1)/(x+1))(2n-1)

(2n-1)

= 2 [ (x-1)/(x+1) + (1/3)( (x-1)/(x+1) )3 + (1/5) ( (x-1)/(x+1) )5 + (1/7) ( (x-1)/(x+1) )7 + ... ]

(x > 0)

Expansions Which Have Logarithm-Based Equivalents

Summantion Expansion Equivalent Value Comments

x nn

= x + (1/2)x2 +(1/3)x3 + (1/4)x4 + ...

= - ln (x + 1) (-1 < x


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For example:

log2(x) dx =x ( log2(x)- 1 / ln(2)) + C

Logarithm approximation

log2(x) n + (x/2n - 1) ,

Logarithms table

x log10x log2x logex

0 undefined undefined undefined0+ - - -

0.0001 -4.000000 -13.287712 -9.2103400.001 -3.000000 -9.965784 -6.907755

0.01 -2.000000 -6.643856 -4.6051700.1 -1.000000 -3.321928 -2.3025851 0.000000 0.000000 0.0000002 0.301030 1.000000 0.6931473 0.477121 1.584963 1.0986124 0.602060 2.000000 1.3862945 0.698970 2.321928 1.6094386 0.778151 2.584963 1.7917597 0.845098 2.807355 1.9459108 0.903090 3.000000 2.079442

9 0.954243 3.169925 2.19722510 1.000000 3.321928 2.30258520 1.301030 4.321928 2.99573230 1.477121 4.906891 3.40119740 1.602060 5.321928 3.68887950 1.698970 5.643856 3.91202360 1.778151 5.906991 4.09434570 1.845098 6.129283 4.24849580 1.903090 6.321928 4.38202790 1.954243 6.491853 4.499810

100 2.000000 6.643856 4.605170200 2.301030 7.643856 5.298317300 2.477121 8.228819 5.703782400 2.602060 8.643856 5.991465500 2.698970 8.965784 6.214608600 2.778151 9.228819 6.396930

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700 2.845098 9.451211 6.551080800 2.903090 9.643856 6.684612900 2.954243 9.813781 6.802395

1000 3.000000 9.965784 6.90775510000 4.000000 13.287712 9.210340

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Logarithm 25 June 08

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