Logarithms
The logarithmic function is the inverse of the exponential function. Logarithms occur frequently in higher math and also in science. They can be used to represent large quantities with small numbers. The logarithm has a base similar to the exponential or power function. The following is the definition of a logarithm.
| If x > 0 and b is a constant and is positive, then | |
| y = logbx if and only if by = x | |
| y is the logarithm | |
| b is the base | |
| x is the argument |
Although the base can be any positive constant, more often it is either 10 or e that is the base. If the base of a logarithm is 10, then it is called a common logarithm. If the base is e then it is called the natural logarithm. The natural logarithm or natural log is used more often than the base 10 log. This is because it has special properties. In addition, it is more frequently written as ln x instead of logex.
The logarithm is an increasing function, if the base is greater than one, logb b > 1, as shown in the following graph.
The following are some of the properties of logarithmic functions:
- The domain is the set of positive real numbers.
- The range is the set of all real numbers.
- The function ln x or logbx passes through the point (1,0)
- The function ln x or logbx has the y-axis as a horizontal asymptote.
Rules of Logarithms
| Master Rule: | y = logbx → by = x | |||||
| ln 1 = 0 | loga1 = 0 | |||||
| ln e = 1 | logaa = 1 | |||||
| ln ex = x | logaan = n | |||||
| eln x = x, x > 0 | alogax = x | |||||
| ln br = r*ln b | logbxr = r*logbx | |||||
| ln xy = ln x + ln y | logb(xy) = logbx + logby | |||||
| ln (x/y) = ln x - ln y | logb(x/y) = logbx - logby | |||||
| ln (1/x) = -ln x | logb(1/x) = -logbx | |||||
| ln n√x = (1/n)ln x | logbn√x = (1/n)logbx | |||||
| ln n√xm = (m/n)ln x | logbn√xm = (m/n)logbx | |||||
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