Answer each of the following. The graph of y= In x shows a point on the graph. Write the logarit

step 1 On this problem, we are given the graph of the natural log of x, and we are also given a point o

Answer each of the following. The graph of y= In x shows a...

Key Concepts

Natural Logarithm

The natural logarithm is a logarithmic function with the base e, where e is an irrational constant approximately equal to 2.71828. It is crucial in many areas of mathematics, including calculus and exponential growth/decay models, and serves as the inverse of the exponential function.

Properties of Logarithmic Functions

Logarithmic functions have unique characteristics such as a restricted domain (only positive real numbers), a continuous and increasing nature, and a curve that passes through a specific point on the graph, typically (1, 0) for the natural logarithm. These properties are essential for understanding how the function behaves and for translating between its graphical and algebraic forms.

Interpreting a Point on a Logarithmic Graph

A point on the graph of a logarithmic function provides a coordinate pair where the x-value is the argument of the logarithm and the y-value is the logarithm of that argument. This relationship is expressed algebraically by equating the y-value to the logarithm of the x-value, which is a direct application of the definition of the logarithmic function.

Expressing the Logarithmic Equation

Using the relationship established by a point on the logarithmic graph, one can write the equation that shows the connection between the logarithm and its argument. If a point with coordinates (a, b) lies on the graph, then the equation b = ln(a) directly encapsulates this relationship, effectively representing the logarithmic function at that specific point.

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