Final answer:
The definition of the logarithmic function requires the base 'b' to be greater than 0 and not equal to 1. It includes the logarithm properties related to products, quotients, and powers. These properties hold true for any base, with natural logarithm and common logarithm as examples.
Step-by-step explanation:
The definition of the logarithmic function for x>0 and b>0, where b≠1, is based on the characteristics of logarithms in relation to exponents and the operation of logarithms on products, quotients, and powers. A logarithmic function is the inverse of an exponential function, with the natural logarithm (ln) being the inverse of the natural exponential function (e to the power of x). For instance, we know that In(ex) = x and eln(x) = x. Similarly, any number can be expressed in terms of e, such as b = eln(b) for a base b that is not 1. The logarithm properties include:
- The logarithm of a product is the sum of the logarithms: log(xy) = log(x) + log(y).
- The logarithm of a quotient is the difference of the logarithms: log(x/y) = log(x) - log(y).
- The logarithm of a number raised to an exponent is the product of the exponent and the logarithm: log(xn) = n log(x).
These principles can be applied to any base logarithm, such as the common logarithm (log) where the base is 10, or the natural logarithm (ln) where the base is e. It is essential that the base b is positive and not equal to 1, which is a requirement for a function to be considered a logarithmic function.