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LogarithmsAndExponentials

# LogarithmsAndExponentials - CSE 1400 Applied Discrete...

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CSE 1400 Applied Discrete Mathematics Logarithms and Exponents Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Logarithms 1 Properties of Logarithms 2 Change of Base Formula 3 Computing the Logarithm of a Number 4 Calculating the Numerals in a Number 4 Problems on Logarithms and Exponentials 5 Abstract Logarithmic functions and their inverses, exponential functions occur in many application areas: annuities and loans, growth and decay, seismic modeling, and algorithm analysis. Logarithms The logarithm base e is called the natural logarithm. Using calculus it can be established that ln ( 1 + x ) x - x 2 2 + x 3 3 - · · · + ( - 1 ) n - 1 x n n where the approximation grows more accurate as the degree n of the approximating MacLaurin polynomial increases. The inverse of the natural logarithm is the exponential base e e x 1 + x + x 2 2! + x 3 3! + · · · + x n n ! MacLaurin’s approximation, together with Horner’s rule provides an algorithm for approximating the logarithmic and exponential functions at a value x = a .

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cse 1400 applied discrete mathematics logarithms and exponents 2 0 1 2 3 4 - 2 - 1 0 1 2 x lg ( x ) - 2 - 1 0 1 2 0 1 2 3 4 x 2 x Properties of Logarithms Logarithms have several useful properties. They were invented to simplify computations. They convert multiplication into addition, division into subtraction, and exponentiation into multiplication. These are the easily established rules. 1 . Log of a product is the sum of logs ln ab = ln a + ln b because e a e b = e a + b 2 . Log of a quotient is the difference of logs ln a b = ln a - ln b
cse 1400 applied discrete mathematics logarithms and exponents 3 because e a e b = e a - b 3 . Log of a power is the power times the log ln a b = b ln a because ( e a ) b = e ab

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