MAT111 epw 11/19/061 Logarithms or Biorhythms of Numbers

MAT111 epw 11/19/062 Logarithms (a review) The logarithm of a number x in base b is the number n such that x = b n and is denoted by The logarithm is the mathematical operation that is the inverse of exponentiation. Remember, exponentiation is raising a number to a power, such as b n = x Although the base b can be any number, frequently used bases are 10 and e ( Euler’s Constant)

MAT111 epw 11/19/063 Logarithms (cont.) Examples log 10 (1000) = 3, because 10 3 = 1000 log 10 (500) = because = 500 Logarithms (logs) to the base 10 are often called common logs. Logs to the base e are called natural logs

MAT111 epw 11/19/064 Rules of Logarithms Taking the logarithm of a power of 10 gives the power log x = x Raising 10 to a power that is the logarithm of a number gives back the number 10 log 10 x = x ( x > 0)

MAT111 epw 11/19/065 Rules of Logarithms (cont.) Remember, powers of 10 are multiplied by adding their exponents, therefore the addition rule for logarithms is: log 10 (xy) = log 10 x + log 10 y (x > 0, y > 0 ) because 10 x · 10 y = 10 x + y

MAT111 epw 11/19/066 Rules of Logarithms (cont.) Remember, powers of 10 are divided by subtracting their exponents, therefore the subtraction rule for logarithms is: log 10 (x/y) = log 10 x - log 10 y (x > 0, y > 0 ) because 10 x  10 y = 10 x - y

MAT111 epw 11/19/067 Rules of Logarithms (cont.) Remember, to raise powers of 10 to other powers, multiply the exponents. Therefore the power rule for logarithms is: log 10 a x = x log 10 a ( a > 0 ) because (10 a ) x = 10 a x

MAT111 epw 11/19/068 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 2 2 = 2  2 = 4 = 3 because 3 3 = 3  3  3 = 27

MAT111 epw 11/19/069 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:

MAT111 epw 11/19/0610 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: log 10 a x = x log 10 a ( a > 0 ) power rule Let x = 1/b, then the power rule becomes the root rule: (a>0, b>0)