Question

How do you find the inflection point of a logarithmic function?

Answer 1

Hint: For a function, an inflection point is a point where the curve changes its shape from concave up to concave down. We can find an inflection point by checking whether its slope at a point has the highest value than any other point. That point is called an inflection point.

Complete step by step answer:
As per the given question, we have to find the inflection points of the logarithmic function.
Let the logarithmic function be $$\ln x$$.
For finding the inflection point of a logarithmic function, we need to take the derivative of the logarithmic function.
We know that the derivative of $$\ln x$$ is $${\displaystyle \frac{1}{x}}$$.
Let $$y=\ln x$$.
So, the first derivative which is denoted as $$y^{\prime }$$ will be$$y^{\prime }={\displaystyle \frac{1}{x}}$$.
we can decide the inflection points based on the second derivative of the function which is given by $$y^{″}={\displaystyle \frac{d}{dx}}(y^{\prime })$$.
 Here, we only require the power rule $${\displaystyle \frac{d}{dx}}(a{x}^n)=(na)x^{n-1}$$.
So, the second derivative which is denoted as $$y^{″}$$ is $$y^{″}={\displaystyle \frac{-1}{x^2}}$$.
If $$y^{″}=0$$ then it is the inflection point. Here, $$y^{″}=0$$ for $$x=\pm \mathrm{\infty }$$. This implies $$\ln x$$ do not have any inflection point.
This implies that $$\ln x$$ is a strictly increasing function.
The graph of $$y=\ln x$$ is as shown below:

Therefore, in this way, we can find the inflection point of any logarithmic function.

Note:
In order to solve these types of problems, we must have enough knowledge about inflection points. We need to know the derivation methods to find the derivative of a function. We should avoid calculation mistakes to get the correct solution. While drawing graphs plot the points wisely to avoid any confusion.