


Help: Logarithmic and exponential functionsHere is a complete list of logarithmic and exponential functions accepted by QuickMath. The tables show the usual form in which the functions appear in textbooks, along with the form accepted by QuickMath. In most cases, the QuickMath version is identical to the textbook version. If there is a function missing which you would like see added to those supported by QuickMath, just send your suggestion to comments@quickmath.com.
PROPERTIES OF LOGARITHMIC FUNCTIONSOBJECTIVES In the previous section we learned that the graph of the inverse function f^{1} 1 is a reflection of the graph of f across the line y = x. Exponential functions are onetoone, so they have inverses. have inverses. Consequently, if we graph the function .f (x) =b^{x} and reflect its graph across the line y = x, the result is the graph of f^{1}. This new function is given the name logarithmic function with base b, and it is written as For example, as Figure 1 shows, the graph of f^{1}(x) = log_{2},
is the reflection of the graph of f^{ }(x) = 2^{x}
across the line y = x. Since logarithms, in a sense, evolve as inversesof
exponential functions, algebra of logarithms is derived from the algebra
of exponents, as we shall see. A logarithm is actually an exponent. For instance, consider the expression
49 = 7^{x} 2 as the logarithm of 49 with base 7, and we write: log_{7 }49 = 2. The logarithm value 2 is the exponent to which 7 is raised to get 49. In
general, we have the following logarithm definition: Let b > 0 and b<> 1. The logarithm of x with base b, which is represented by y, is defined by y = log_{b }x if and only if x = b^{y }for every x > 0
and for every real number y Base of logarithm is the same as exponent base Convert each exponential form equation to logarithmic form, The equivalencies are listed in the following table. Convert each logarithmic form to an equivalent exponential form. At times, we can find the numerical value of a logarithm by converting to
exponential form and then using the onetoone property of exponents, as the
next example illustrates. Evaluate each logarithm. In each case we let u equal the given expression, and write the logarithmic equation in its equivalent exponential form and then solve the resulting equation for u as shown in the following table.
Solve each logarithmic equation for x. First we rewrite the given equation in exponential form and then solve the resulting equation. Evaluating Logarithms  Base 10 and Base e The base of a logarithmic function can be any positive number except 1. However, the two bases that are most widely used are l0 and e.A logarithm with base 10 is called a common logarithm. Its value at x is denoted by log x, that is, A logarithm with base e is called a natural logarithm, and its value at x is denoted by In x, that is, Consequently:
