# Graphical Transformation of y=ln (x)

• AN630078
In summary, y=ln(2x) is a horizontal stretch of y=ln(x) with a scale factor of 1/2, while y=ln(4-x) is a reflection in the y-axis followed by a translation of the graph line to the left by 4 units.
AN630078
Homework Statement
Hello, I was practising describing graphical transformations with several example questions but there was one which I was especially unsure of. I would appreciate any advice upon my proposed solutions.
Describe the transformation which maps y=ln(x) to;
a. y=ln(2x)
b.y=ln(4-x)
Relevant Equations
y=f(ax) is a horizontal stretch by a scale factor 1/a
y=f(-x) is a reflection in the y-axis
y=f(x+a) is a translation by the vector (-a,0)
a. I believe that y=ln(2x) is a horizontal stretch of y=ln(x) of scale factor 1/2. In the transformation y=ln(2x), each x-value is multiplied by 2 before the corresponding y-value is calculated.

b. I think that y=ln(4-x) is a reflection in the y-axis followed by a translation by the vector (-4,0) i.e. 4 units to the right.

Just to be clear, you need to say if you are talking about the x-axis being stretched and reflected or about the graph line.

hutchphd
AN630078 said:
b. I think that y=ln(4-x) is a reflection in the y-axis followed by a translation by the vector (-4,0) i.e. 4 units to the right.
Try reducing that to one operation.

FactChecker said:
Just to be clear, you need to say if you are talking about the x-axis being stretched and reflected or about the graph line.
If we take the relevant equations as the guide, it must mean a transformation of the curve, keeping the coordinates fixed.

I would not say that a translation"by the vector (-4,0)" is "4 units to the right." I would say that it is moving the points of the graph line to the left versus the axis system.

SammyS

## 1. What is the general transformation rule for y=ln(x)?

The general transformation rule for y=ln(x) is that any changes made to the argument (x) will result in the same changes being made to the output (y). For example, if the argument is multiplied by a constant, the output will also be multiplied by the same constant.

## 2. How does changing the base of the logarithm affect the graph of y=ln(x)?

Changing the base of the logarithm will result in a vertical stretch or compression of the graph. A larger base will result in a steeper graph, while a smaller base will result in a flatter graph.

## 3. What is the domain and range of y=ln(x)?

The domain of y=ln(x) is all positive real numbers, since the natural logarithm is only defined for positive numbers. The range is all real numbers, as the output can be any real number depending on the input.

## 4. How does adding or subtracting a constant affect the graph of y=ln(x)?

Adding or subtracting a constant to the argument (x) will result in a horizontal shift of the graph. Adding a constant will shift the graph to the left, while subtracting a constant will shift it to the right.

## 5. What is the asymptote of the graph of y=ln(x)?

The asymptote of y=ln(x) is the y-axis, which the graph approaches but never intersects. This is because the natural logarithm is undefined for x=0.

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