Transcript The Decimal Number System

MAT111
Logarithms
or
Biorhythms of Numbers
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MAT111
Logarithms (a review)
The logarithm of a number x in base b is the number n
such that x = bn and is denoted by
logb ( x )  n
The logarithm is the mathematical operation that is the
inverse of exponentiation. Remember, exponentiation is
raising a number to a power, such as bn = x
Although the base b can be any number, frequently used
bases are 10 and e (Euler’s Constant)
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MAT111
Logarithms (cont.)
Examples
log10(1000) = 3, because 103 = 1000
log10(500) = 2.6989700043360188047863
because 102.6989700043360188047863 = 500
Logarithms (logs) to the base 10 are often
called common logs.
Logs to the base e are called natural logs
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MAT111
Rules of Logarithms
• Taking the logarithm of a power of 10
gives the power
log1010x = x
• Raising 10 to a power that is the logarithm
of a number gives back the number
log10x
10
=x
(x > 0)
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MAT111
Rules of Logarithms (cont.)
• Remember, powers of 10 are multiplied by
adding their exponents, therefore the
addition rule for logarithms is:
log10(xy) = log10x + log10y (x > 0, y > 0)
because
10x · 10y = 10x+y
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MAT111
Rules of Logarithms (cont.)
• Remember, powers of 10 are divided by
subtracting their exponents, therefore the
subtraction rule for logarithms is:
log10(x/y) = log10x - log10y (x > 0, y > 0)
because
10x  10y = 10x-y
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MAT111
Rules of Logarithms (cont.)
• Remember, to raise powers of 10 to other
powers, multiply the exponents. Therefore
the power rule for logarithms is:
log10ax = x log10a (a > 0)
because
x
a
(10 )
= 10ax
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MAT111
Roots (a slight digression)
• Finding a root is the reverse of raising a
number to a power
• We indicate an nth root of a number by
writing the number under the n symbol
• Examples
4  2 because 22 = 22 = 4
3
27 = 3 because 33 = 333 = 27
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MAT111
Roots (a slight digression)
• We indicate an nth root of a number by
writing the number under the n symbol
• The nth root of a number is the same as
the number raised to the 1/n power:
n
x x
1
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MAT111
Rules of Logarithms (cont.)
• Remember, the nth root of a number is the
same as the number raised to the 1/n
power. Therefore we use the power rule
for logs to produce the root rule for logs:
log10ax = x log10a (a > 0) power rule
Let x = 1/b, then the power rule becomes
the root rule:
1
log10 a
b
b
(a>0, b>0)
log10 ( a)  log10 (a ) 
b
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