How to Graph a Logarithmic Function with 4 types of Transformations | Extraclass

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This video will help you learn : How to graph a logarithmic function?
Before learning this let us first understand.
Why do we need to learn to graph any function?
Practically drawing a rough sketch of a function aids in remembering the definition and properties with visualizations rather than mugging them up.
Now since every logarithmic function of the form y equal to log of x to base b is the inverse of an exponential function with the form y equal to b raised to the power x.
Hence, their graphs will be reflections of each other across the line y equals to x.
So if we have two graphs of f of x equal to 2 raised to the power x and G of x equal to log of x base 2, we can see that the X and Y coordinates are reversed in either graphs.
COMPARISON OF GRAPHS :
We can compare the two graphs based on the following points.
1. Intercept:
Y intercept of f of x at 0 comma 1 is the reflection of x intercept of G of X at 1 comma 0.
2. Domain and Range:
The domain of f of x is the same as the range of G of X and varies from minus infinity to plus infinity. The range of f of x is the same as the domain of G of X and varies from 0 to Infinity.
Now if we consider the parent logarithmic function f of x equal to log of x to base b only for any real number X and base be greater than 0 and not equal to 1 we can observe the following characteristics from the graph.
It is a one-to-one function meaning every value of x gives a distinct and unique value of y. Domain of the function vary from 0 to infinity and range varies from minus infinity to plus infinity.
The graph has x intercept one and No Y intercept with a key point at B comma 1.
Moreover the graph is increasing when B is greater than 1 and decreasing if b is less than 1 with both graphs having an asymptote at x equal to 0 meaning both the Graph’s coverage towards Infinity at x equal to zero.
TRANSFORMATION OF GRAPHS
Now let's move on to learn the method of transforming logarithmic graph starting from the parent function.
Like most functions we can shift, stretch, compress and reflect the parent function without loss of shape.
First Transformation : Horizontal Shift
When a constant C is added to the input of the parent function f of x equal to log of x to base B. The result is a horizontal shift c units in the opposite direction of the sign of C.
When a constant C greater than 0 is added to the input of the parent function the graph is shifted along side left and when C is subtracted from the input the graph is shifted along side right.
Second Transformation : Vertical Shift
When a constant D greater than 0 is added or subtracted to the parent function the result is a vertical shift d units in the direction of the sign on D. When D is added to the parent function the graph is vertically shifted up by D units and When D is subtracted from the parent function the graph is vertically shifted down by D units.
Third Transformation : Stretch and Compression
When the parent function is multiplied by a constant the result is a vertical stretch or compression of the original graph. When a constant A say greater than 1 is Multiplied to the parent function the transformation is a vertical stretch. Similarly when the parent function is divided by the same constant the transformation is a vertical compression.
Fourth Transformation : Reflections
When the parent function is multiplied by x -1 the result is a reflection about the x-axis and when the input is multiplied by x -1, the result is a reflection about the y axis.
First let us restrict the base B to be greater than 1. Now we can see that the graph undergoes reflection about the x-axis when the parent function is multiplied by -1. Similarly the graph will undergo reflection about the y axis when the input is multiplied by -1 .
Thus we can observe that in both the cases the new transform function changes its monotonicity that from being monotonically increasing it transforms into a monotonically decreasing graph.
Keeping the visual changes of all these four transformations in mind will help us in graphing any logarithmic function.
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[@prabhatsaini994]

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[@parvejkhan-py8sg]

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[@Sarvankumar-df1fl]

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