How is the graph of log (x – 5) translated from the graph of log x?
no matter what the function
f(x) ---> f(x-5) results in a horizontal translation of 5 units to the right
look at the graph
http://www.wolframalpha.com/input/?i=plot+y+%3D+log%28x%29+%2C+y+%3D+log%28x-5%29+for+0%3C+x%3C30
To understand how the graph of log(x - 5) is translated from the graph of log(x), we need to understand the concept of translation.
Translation is a transformation that shifts a graph horizontally and/or vertically. In this case, we are interested in the horizontal shift of the graph.
The general form of a logarithmic function is y = logb(x), where b is the base of the logarithm and x is the input value. For simplicity, let's consider the natural logarithm, where the base is e. So, y = ln(x).
When we have a transformation of the form log(x - a), where "a" is a constant, it implies a horizontal shift to the right by "a" units. In this case, log(x - 5) suggests a shift to the right by 5 units, compared to the graph of log(x).
To illustrate this, let's consider a specific example. We can compare the graph of log(x) to the graph of log(x - 5) by plotting some points.
For the original graph of log(x), we can choose a few x-values and calculate the corresponding y-values using the natural logarithm:
For x = 1, y = ln(1) ≈ 0
For x = 2, y = ln(2) ≈ 0.693
For x = 3, y = ln(3) ≈ 1.099
Now, for the shifted graph of log(x - 5), we can plug in x - 5 into the natural logarithm:
For x = 6, y = ln(6 - 5) = ln(1) ≈ 0
For x = 7, y = ln(7 - 5) = ln(2) ≈ 0.693
For x = 8, y = ln(8 - 5) = ln(3) ≈ 1.099
If we plot these points on the coordinate plane, we will see that the graph of log(x - 5) is shifted to the right by 5 units compared to the graph of log(x).
So, the graph of log(x - 5) is a translation of the graph of log(x) to the right by 5 units.