Understanding ln |x| + C: Properties and Graphical Representation of Natural Logarithm of Absolute Value Function with a Constant C

ln |x| + C

∫1/x dx

The expression ln |x| + C represents the natural logarithm of the absolute value of x, plus a constant C. Here are a few key points about this expression:

– The natural logarithm (ln) is a mathematical function that returns the logarithm of a number with respect to the mathematical constant e (approximately equal to 2.718).
– The absolute value of x (|x|) is the distance of x from zero on the number line, regardless of whether x is positive or negative.
– Adding a constant C to the natural logarithm does not change the shape of the graph of the function, but it shifts the entire graph vertically by a distance of C units.

To fully understand the behavior of the expression ln |x| + C, it is useful to graph it. The graph of ln |x| + C is a curve that passes through the point (1, C) on the y-axis, and the asymptotes x = 0 and y = -infinity.

Some important properties of ln |x| + C:

– The domain of the function is all real numbers except x = 0.
– As x approaches 0 from the left or right, the function approaches negative infinity.
– As x becomes very large in the positive or negative direction, the function increases without bound.
– The derivative of ln |x| + C is 1/x. This means that the slope of the tangent line to the graph at any point (x, ln |x| + C) is 1/x.

In summary, ln |x| + C represents the natural logarithm of the absolute value of x, plus a constant C. The domain of the function is all real numbers except x = 0, and the graph has asymptotes x = 0 and y = -infinity. The derivative of the function is 1/x.

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