Exponential and Logarithmic Functions


Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = a^x is x = a^y The logarithmic function y = logax is defined to be equivalent to the exponential equation x = a^y. y = logax only under the following conditions: x = a^y, a > 0, and a≠1. It is called the logarithmic function with base a. Consider what the inverse of the exponential function means: x = a^y. Given a number x and a base a, to what power y must a be raised to equal x? This unknown exponent, y, equals logax. So you see a logarithm is nothing more than an exponent. By definition, alogax = x, for every real x > 0. Here are some useful properties of logarithms, which all follow from identities involving exponents and the definition of the logarithm. Remember a > 0, and x > 0. loga1 = 0 , logaa = 1 , loga(a^x) = x , alogax^ = x , loga(bc) = logab + logac , loga(b/c) = logab - logac , loga(x^d) = d logax. An exponential function is a function in which the independent variable is an exponent. Exponential functions have the general form y = f (x) = a^x, where a > 0, a≠1, and x is any real number. The reason a > 0 is that if it is negative, the function is undefined for -1 < x < 1. Restricting a to positive values allows the function to have a domain of all real numbers. In this example, a is called the base of the exponential function. Here is a little review of exponents: a^-x = 1/a^x , a^x+y = a^x.a^y , a^x-y =a^x/a^y , a^0 = 1.



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