Download MATH 241 Lecture 6: Logarithmic Functions and Growth of Exponentials - Prof. Michael Price and more Study notes Mathematics in PDF only on Docsity! MATH 241, LECTURE 6 1. Logarithmic functions Many common functions arise through “undoing” basic functions (more formally, we say they are inverses of basic functions). For example, subtraction was invented as the inverse of addition, division as the inverse of multiplication, and the square root as the inverse of the squaring function. Definition 1. The inverse of the exponential function ax is called the logarithm function (with a base of a) denoted loga(x). By this definition, loga ax = x and aloga x = x. The logarithm with a base of e is called the natural log function and is denoted ln(x). Example 2. • log10 10000 = 6 because 106 = 100000. • log2 18 = −3 because 2 −3 = 18 . • ln √ e = 12 because e 1 2 = √ e. The properties of the logarithm function follow from those of the exponential function. • loga(xy) = loga x + loga y because ax+y = axay. • loga xy = y loga x because (ax)y = axy. • logb x = logb a · loga x follows from the previous property. In other words, logarithms turn multiplication to addition and turn exponentiation to multiplication. Also note that the last property says that logarithms with different bases are related by multiplication by a constant. Logarithms are handy in problems involving exponentials. Example 3. A radioactive material decays at a rate of 0.5% per year. How long would it take for half of the material to decay? Example 4. How long would it take for $1000 invested at 7% to become $1500? How long would it take the balance to double? 2. Growth and graphs of exponentials and logarithms Exponential functions grow (or decay) very quickly. Logarithmic functions grow extremely slowly. For example, we may look at how fast each of these functions “go from 1 to 100:” x; x2; 3x; log10 x. Technology permitting, we may elaborate on this by looking at graphs of these and other functions. 3. Average rate of change A fundamental philosophical truth is that everything changes. In physics, the change in position is known as velocity or speed. In economics, the change in price is known as inflation. In business, the change in costs is sometimes known as trend. In mathematics, the change in values of a function is known as the derivative. But to understand the derivative, which will measure “instantaneous” change, you need to to first be comfortable with “average” change over some intervals. 1
