Final answer:
The logarithm of x with base b, log_b(x), represents the exponent to which b must be raised to yield x. Logarithm and exponentiation are inverse operations. For a number raised to an exponent, the logarithm is the product of the exponent and the logarithm of the base number.
Step-by-step explanation:
The logarithm of x with base b, written as logb(x), is the power to which the base b must be raised to produce the number x. This is directly related to the operation of b raised to the logarithm with base b of a number, because if you take the base b and raise it to logb(y), where y is any number, you get back the original number y. This is due to the fact that logarithm and exponentiation are inverse operations.
Moreover, if you have a number raised to an exponent, the logarithm of that number with any base is the product of the exponent and the logarithm of the number without the exponent. This rule is reflected in the identity: logb(xn) = n · logb(x), which is especially helpful in simplifying complex expressions.
There are various types of logarithms, such as the common logarithm, log10(x), which uses 10 as its base, or the natural logarithm, ln(x), which uses e (approximately 2.7182818) as its base. It's important to note that regardless of the base, the properties of logarithms are uniform and apply to all logarithmic forms.