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MAT111 Logarithms or Biorhythms of Numbers epw 11/19/06 1 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) epw 11/19/06 2 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 epw 11/19/06 3 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) epw 11/19/06 4 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 epw 11/19/06 5 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 epw 11/19/06 6 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 epw 11/19/06 7 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 = 22 = 4 3 27 = 3 because 33 = 333 = 27 epw 11/19/06 8 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 epw 11/19/06 n 9 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 epw 11/19/06 10