Mathematics Dictionary Dr. K. G. Shih

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Logarithm

Subjects

Read Symbol defintion Q01 | - Defintion Q02 | - Laws of logarithm Q03 | - Inverse function of logarithm Q04 | - Common logarithm Q05 | - Why is common logarthm called Briggs logaithm Q06 | - Natural logarithm y = ln(x) Q07 | - Find ln(2) Q08 | - Find ln(3) Q09 | - Use ln(2) to find ln(x) Q10 | - Use ln(2) and ln(3) to find ln(x) Q11 | - Use log10(2) and log10(3) to find log10(x) Q12 | - The logarithmic family functions Q13 | - Find log4(64) where log4(64) is log(64) to base 4

Answers

Q01. Definition If 10^x = b, then log10(b) = x. Where log10(b) is log(b) to the base 10. Since 10^1 = 10, hence log10(10) = 1. Since 10^0 = 1, hence log10(1) = 0. Symbol on computer ^ means power. Example 2^3 = 8. * means multiplication. Example 2*3 = 6. e^x means e to power x. log5(x) means log(x) to base 5. Go to Begin Q02. Formula Log(A*B) = Log(A) + Log(B) Log(A/B) = Log(A) - Log(B) Log(A^n) = n*Log(A) Log(1) = 0 Natural logarithm It is ln(x) with base e. ln(e) = 1. if y = ln(x) then x = e^y. Change base formula Log10(x) = ln(x)/ln(10) Example : Use calculator find log5(10) which is log(1) to base 5. Method 1 : log5(10) = ln(10)/ln(5) = 2.302585/1.609438 = 1.430676 Method 2 : log5(10) = log10(10)/log10(5) = 1/0.698970 = 1.430676 Go to Begin Q03. Inverse of logarithm If y = e^x then y = ln(x) is the inverse Properties of inverse Property 1 : ln(e^x) = x Property 2 : e^(ln(x)) = x What is inverse ? Use line y = x as an mirror. y = e^x is the image of y = ln(x) relative with y = x. y = ln(x) is the image of y = e^x relative with y = x. Diagrams of y = e^x and y = ln(x) Program 06 05 Go to Begin Q04. Common logarithm It is also called Briggs logaithm. Log10(x) is log(10) to base 10. Special values log10(0.0001) = -4 log10(0.001) = -3 log10(0.01) = -2 log10(0.1) = -1 log10(1) = 0 log10(10) = 1 log10(100) = 2 log10(1000) = 3 log10(10000) = 4 Go to Begin Q05. Why is common logarthm called Briggs logaithm Henry Briggs (1561-1639) prepared the common logarithmic table. The Briggs logarithm table covers numbers from 1 to 1000 accurate to 14 places. The methods prepare the table by Briggs : Examples Example 1 Since Sqr(10) = 3.16227...., Hence log10(3.162277...) = 0.5 Example 2 10^(3/4) = Sqr(31.62277...) = 5.623413..., Hence log10(5.623413...) = 0.75 Example 3 log10(5.623413...) + log10(3.162277...) = 0.75 + 0.25 log10(17.782794...) = 1.25 Go to Begin Q06. Natural logarithm y = ln(x) Facts It is also called Napirian logarithm or hyperbolic logarithm. Natural logarithm is expresses as ln(x). The base is e = 2.718.... Why did it call Napirian logarithm ? Since the logarithm function implies in Napir's defintion in logarithm. y = ln(x) Equivalent exponent : x = e^y. Derivative : y' = 1/x Series : ln(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + .... Series : ln(1-x) = -(x + (x^2)/2 + (x^3)/3 + (x^4)/4 + ....) Inverse of y = ln(x) is y = e^x, e^(ln(x)) = x ln(e^x) = x Properties of y = ln(x) x must be greater than 0. Hence the curve has no y-intercept. x = 1 and y = 0. x = 0 and y = -infinite. x = e and y = 1. The curve is always increasing. The curve is concave downward. Go to Begin Q07. Find ln(2) Calculator method ln(2) = 0.6931471.... Series method : Use ln(1+x) to find ln(2) Use series of ln(1+x) and let x = 1 ln(2) = 1 - 1/2 + 1/3 - 1/4 + 1/5 - .... ln(2) = 1 - 0.5 + 0.33333 - 0.25 + 0.2 - 0.16667 + ... Converge very slow Series method : Use ln(1-x) to find ln(2) Use series of ln(1-x) and let x = 0.5 ln(1/2) = -(1/2 + (1/2)^2/2 + (1/2)^3/3 + (1/2)^4/4 + ....) ln(1/2) = -(0.5 + 0.125 + 0.0416666 + 0.015625 + 0,00625 + 0.0026041 +...) ln(1/2) = -0.6915057 Since ln(1/2) = ln(1) - ln(2) and ln(1) = 0 Hence ln(2) = 0.6915057 Discussion It only accurate to 2 decimal place by using first 7 terms. This is much better than using series of ln(1+x) From this example we see that the calculation of ln(x) is not easy. Hence we need ln(x) table before we can use the calculator. Go to Begin Q08. Find ln(3) Calculator method ln(3) = 1.098612289 Use seires of ln(1+x) We use series of ln(1+x) and let x = 0.5 ln(3/2) = 1/2 - (1/2)^2/2 + (1/2)^3/3 - (1/2)^4/4 + ....) ln(3/2) = 0.5 - 0.125 + 0.0416666 - 0.015625 + 0.00625 - 0.0026041 +...) Hence ln(3/2) = 0.3211543... Since ln(3/2) = ln(3) - ln(2) = 0.3211543 ln(3) = 0.6915057 + 0.3211543 = 1.01266 Go to Begin Q09. Use ln(2) to find ln(x) Find ln(4) ln(2) = 0.69314718 ln(4) = ln(2^2) = 2*ln(2) = 1.38628436 Find ln(8) ln(2) = 0.69314718 ln(8) = ln(2^3) = 3*ln(2) = 2.07944154 Find ln(16) ln(2) = 0.69314718 ln(16) = ln(2^4) = 4*ln(2) = 2.77258872 Go to Begin Q10. Use ln(2) and ln(3) to find ln(x) Find ln(6) ln(2) = 0.69314718 and ln(3) =1.09861229 ln(6) = ln(2*3) = ln(2) + log(3) = 1.79175957 Find ln(1.5) ln(2) = 0.69314718 and ln(3) =1.09861229 ln(1.5) = ln(3/2) = ln(3) - ln(2) = 0.40546511 Go to Begin Q11. Use log10(2) and log10(3) to find log10(x) Given that log10(2) = 0.30103 and log10(3) = 0.47712 Find log10(1.5) Log10(1.5) = log10(3/2) = log10(3) - log10(2) log10(1.5) = 0.17609 Find log10(6) Log10(6) = log10(3) + log10(2) log10(6) = 0.77815 Find log10(9) Log10(9) = log10(3^2) = 2*log10(3) log10(9) = 1.43136 Go to Begin Q12. The logarithmic family functions Inverse hyperbolic functions arcsinh(x) = arccosh(x) = arctanh(x) = arccsch(x) = arcsech(x) = arccoth(x) = Reference Study subject Properties of inverse hyperbolic functions Study subject Properties of exponent Go to Begin Q13. Find log4(64) which is log(64) to base 4 Use log(x^m) = m*log(x) log4(64) = log4(4^3) log4(64) = 3*log4(4) = 3 By defintion Let Log4(64) = x, then 64 = 4^x. 64 = 4^3 and hence x = 3 Use ln(x) on calculator Change base log4(64) = ln(64)/ln(4) = 4.15888308/1.38629436 = 3 Go to Begin

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