Question

How would you graph $y = \ln \left( {x - 1} \right) + 3$ ?

Answer 1

Hint : A graph of a function f is the set of ordered pairs; the equation of graph is generally represented as $y = f\left( x \right)$ , where x and $f\left( x \right)$ are real numbers. We substitute the value of x and we determine the value of y and then we mark the points in the graph and we join the points.

Complete step by step solution:
Here, in the given question, we have to plot the graph for the given function. A graph of a function is a set of ordered pairs and it is represented as $y = f\left( x \right)$ , where x and $f\left( x \right)$ are real numbers. These pairs are in the form of cartesian coordinates and the graph is the two-dimensional graph.
First, we have to find the value of y by using the graph equation $y = \ln \left( {x - 1} \right) + 3$ .
Let us substitute the value of x as $2$ .
$\Rightarrow y = \ln \left( {2 - 1} \right) + 3$
$\Rightarrow y = \ln \left( 1 \right) + 3$
We know that the value of $\ln \left( 1 \right)$ is zero. So, we get,
$\Rightarrow y = 0 + 3$
$\Rightarrow y = 3$
Now we consider the value of x as $e + 1$ , the value of y is
$\Rightarrow y = \ln \left( {\left( {e + 1} \right) - 1} \right) + 3$
Simplifying the expression,
$\Rightarrow y = \ln \left( e \right) + 3$
Now, we know that the value of $\ln \left( e \right)$ is one. So, we get,
$\Rightarrow y = 1 + 3$
$\Rightarrow y = 4$
Now we consider the value of x as ${e^2} + 1$ , the value of y is
$\Rightarrow y = \ln \left( {\left( {{e^2} + 1} \right) - 1} \right) + 3$
Simplifying the expression,
$\Rightarrow y = \ln \left( {{e^2}} \right) + 3$
Now, we know that the value of $\ln \left( e \right)$ is one. So, we get,
$\Rightarrow y = 2 + 3$
$\Rightarrow y = 5$
Now we draw a table for these values we have

x	$2$	$\left( {e + 1} \right)$	$\left( {{e^2} + 1} \right)$
y	$3$	$4$	$5$

We also know the nature of the graph of logarithmic function. Hence, we can now plot the graph of the given function $y = \ln \left( {x - 1} \right) + 3$ with the help of coordinates of the points. The nature of the graph of a function and its slope can also be determined from the derivative of the function. The graph plotted for these points is represented below:

Note : The number ‘e’ in mathematics is known as the euler’s number. It is a mathematical constant and has a value approximately equal to $2.71828$ . It is the base of the natural logarithm that is used in many steps in the given problem.