A Few Words About Logarithms and Exponentials Help
Introduction to a A Few Words About Logarithms and Exponentials
We will give a more thorough treatment of the logarithm and exponential functions in Chapter 6. For the moment we record a few simple facts so that we may use these functions in the sections that immediately follow.
The logarithm is a function that is characterized by the property that
log( x · y ) = log x + log y .
It follows from this property that
log( x / y ) = log x − log y
log( x n ) = n · log x.
It is useful to think of log a b as the power to which we raise a to get b , for any a , b > 0. For example, log 2 8 = 3 and log 3 (1/27) = −3. This introduces the idea of the logarithm to a base.
You Try It: Calculate log 5 125, log 3 (1/81), log 2 16.
The most important base for the logarithm is Euler’s number e ≈ 2.71828 .... Then we write ln x = log e x . For the moment we take the logarithm to the base e , or the natural logarithm , to be given. It is characterized among all logarithm functions by the fact that its graph has tangent line with slope 1 at x = 1. See Fig. 1.58. Then we set
Note that this formula gives immediately that log e x = In x , once we accept that log e e = 1.
Math Note: In mathematics, we commonly write log x to mean the natural logarithm. Thus you will sometimes encounter In x and sometimes encounter log x (without any subscript); they are both understood to mean log e x , the natural logarithm.
The exponential function exp x is defined to be the inverse function to In x. Figure 1.59 shows the graph of y = exp x . In fact we will see later that exp x = e x. More generally, the function a x is the inverse function to log a x. The exponential has these properties:
(a) a b + c = a b · a c;
These are really just restatements of properties of the logarithm function that we have already considered.
You Try It: Simplify the expressions 3 2 · 5 4 /(15) 3 and 2 4 · 6 3 · 12 −4.
Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.
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