MAT111

epw 11/19/06 1

Logarithms

or

Biorhythms of Numbers

MAT111

epw 11/19/06 2

Logarithms (a review)

The logarithm of a number x in base b is the number n such that x = bn and is denoted by

               

The logarithm is the mathematical operation that is the inverse of exponentiation. Remember, exponentiation israising 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)

log ( ) nx b

MAT111

epw 11/19/06 3

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

MAT111

epw 11/19/06 4

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

10log10x = x (x > 0)

MAT111

epw 11/19/06 5

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

MAT111

epw 11/19/06 6

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

MAT111

epw 11/19/06 7

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

(10a)x = 10ax

MAT111

epw 11/19/06 8

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 symbol

• Examples

because 22 = 22 = 4

= 3 because 33 = 333 = 27

n

4 2

3 27

MAT111

epw 11/19/06 9

Roots (a slight digression)

• We indicate an nth root of a number by writing the number under the symbol

• The nth root of a number is the same as the number raised to the 1/n power:

n

1x xn n

MAT111

epw 11/19/06 10

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:

(a>0, b>0)10

10 10

1 log alog ( a) log (a )

b b b