lnx monotonicity
monotonicity of ln(x) is equal to sign of (ln(x))'=1/x,
and domain of ln(x) is D=(0,+∞)
Since 1/x is strictly positive for x∈D, so ln(x) is strictly monotonically increasing.
*...is related to sign of...
The term "lnx monotonicity" refers to the study of the monotonicity properties of the natural logarithmic function, often denoted as ln(x).
To understand the monotonicity of ln(x), we need to explore how the logarithmic function behaves as its input, x, varies.
First, let's define what monotonicity means. A function is said to be monotonic if it consistently maintains a specific order as its input values change. There are two types of monotonicity: increasing and decreasing.
In the case of ln(x), we focus on its increasing monotonicity. This means that as the input value, x, increases, the corresponding value of ln(x) also increases. In other words, ln(x) becomes larger as x gets larger.
To prove the increasing monotonicity of ln(x), we can use calculus. The derivative of ln(x) with respect to x is given by 1/x. The derivative is positive for all positive values of x, indicating that ln(x) is an increasing function. Consequently, ln(x) is strictly increasing as x grows.
Regarding the decreasing monotonicity of ln(x), we need to note that ln(x) is only defined for positive values of x. Therefore, it doesn't possess a decreasing monotonicity because it is not defined for negative values or zero.
In summary, the natural logarithmic function, ln(x), is an increasing function for positive values of x. This means that as x increases, ln(x) also increases.