Mathematics Dictionary
Dr. K. G. Shih
Logarithm
Subjects
Symbol Defintion
Example : Sqr(x) = Square root of x
AL 09 00 |
- Outlines
AL 09 01 |
- Defintion
AL 09 02 |
- Laws of logarithm
AL 09 03 |
- Inverse function of logarithm
AL 09 04 |
- Common logarithm (base 10)
AL 09 05 |
- Why is common logarthm called Briggs logaithm
AL 09 06 |
- Natural logarithm y = ln(x)
AL 09 07 |
- Find ln(2)
AL 09 08 |
- Find ln(3)
AL 09 09 |
- Use ln(2) to find ln(x)
AL 09 10 |
- Use ln(2) and ln(3) to find ln(x)
AL 09 11 |
- Use log10(2) and log10(3) to find log10(x)
AL 09 12 |
- The logarithmic family functions
AL 09 13 |
- Find log2(64) where log2(64) is log(64) to base 2
AL 09 14 |
- Find log3(81) where log3(81) is log(81) to base 3
AL 09 15 |
- Find log4(64) where log4(64) is log(64) to base 4
Answers
AL 09 01. 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.
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AL 09 02. Formula
Rules
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
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AL 09 03. 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
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AL 09 04. 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
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AL 09 05. 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
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AL 09 06. 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.
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AL 09 07. 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 the calculator is available.
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AL 09 08. 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
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AL 09 09. 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
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AL 09 10. 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
Find ln(0.75)
ln(1.5/2) = ln(1.5) - ln(2)
= 0.40546511 - 0.69314718 = -0.28768207
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AL 09 11. Use log10(2) and log10(3) to find log10(x)
Question : 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
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AL 09 12. 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
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AL 09 13. Find log2(64) which is log(64) to base 2
Use log(x^m) = m*log(x)
log2(64) = log4(2^6)
log2(64) = 6*log2(2) = 4
By defintion
Let Log4(64) = x, then 64 = 2^x.
64 = 2^6 and hence x = 6
Use ln(x) on calculator
Change base formula
log2(64) = ln(64)/ln(4) = 4.15888308/0.693147181 = 6
or
log2(64) = ln(64)/ln(2) = ln(2^6)/ln(2) = 6*ln(2)/ln(2) = 6
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AL 09 14. Find log3(81) which is log(81) to base 3
Use log(x^m) = m*log(x)
log3(81) = log3(3^4)
log3(81) = 4*log3(3) = 4
By defintion
Let Log3(81) = x, then 81 = 3^x.
81 = 3^4 and hence x = 4
Use ln(x) on calculator
Change base formula
log3(81) = ln(81)/ln(3) = 4.394449155/1.098612289 = 4
or
log3(81) = ln(81)/ln(3) = ln(3^4)/ln(3) = 4*ln(3)/ln(3) = 4
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AL 09 15. 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 formula
log4(64) = ln(64)/ln(4) = 4.15888308/1.38629436 = 3
log4(64) = ln(64)/ln(4) = ln(4^3)/ln(4) = 3*ln(4)/ln(4) = 3
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AL 09 00. Outlines
Rules
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)
Inverse of ln(x)
Inverse of ln(x) is e^x and ln(e^x) = x
Inverse of e^x = ln(x) and e^(ln(x)) = x
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