Question

What is the function f(x) and g(x)?

1) f(x) = log(x² - 1) and g(x) = log(x)
2) f(x) = log(x² + 1) and g(x) = log(x)
3) f(x) = log(x² - 1) and g(x) = log(x²)
4) f(x) = log(x² + 1) and g(x) = log(x²)

Answer 1

The student's question involves understanding the properties of logarithmic functions and their interactions with exponential functions. We apply logarithmic properties to express and simplify mathematical expressions.

Step-by-step explanation:

The functions f(x) and g(x) are provided in various forms involving the logarithmic function. By applying the properties of logarithms, such as the property that the logarithm of a product of two numbers is the sum of the logarithms of those numbers, and that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, we can analyze these functions.

For instance, understanding that the natural logarithm and the exponential are inverse functions allows us to greatly simplify our computations with logarithms, as they will "undo" each other.