Final answer:
To determine the domain of a logarithmic function, identify that the function is defined only for positive real numbers. This is reflected in the fundamental properties of logarithms which maintain positivity within the logarithm operation. The domain excludes zero and negative numbers, so it is the set of all positive real numbers.
Step-by-step explanation:
Determining the Domain of a Logarithmic Function
To determine the domain of a logarithmic function, you must first understand what the function looks like and the basic properties of logarithms. A logarithmic function is the inverse of an exponential function, and it is defined only for positive real numbers. This is because the logarithm of a non-positive number is undefined in real numbers. Hence, the domain of a logarithmic function includes all positive real numbers.
Let's consider one of the fundamental properties of logarithms that states: the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers, mathematically expressed as log(a/b) = log(a) - log(b). This property applies to any base, including common logarithms (log base 10) and natural logarithms (ln, which is log base e). These operations within the logarithm must result in a positive outcome for the function to be valid, as you cannot take the logarithm of zero or a negative number.
The domain of any logarithmic function logb(x) will exclude zero and negative numbers, so the domain is all positive real numbers, i.e., x > 0. Although simplification of the function might sometimes make it appear as if the domain could include non-positive numbers, it is essential to remember that the domain is restricted by the initial definition of the logarithm, which only applies to positive numbers.
In the context of plotting, especially in scientific fields such as chemistry and biology, logarithmic functions are useful in dealing with a wide range of values. A logarithmic scale enables the comparison of values that differ greatly in magnitude.