I love your blog entry on the use of a semilog graph to emphasize the area of greatest interest in a ranking situation and will be using it in my precalculus class a week or so from now.
I can also imagine that a loglog graph might be useful to get a view of high frequency small amplitude oscillations near the origin and/or big slow variations far away, but would be hard put to come up with such a topical example.
Of course a loglog graph is also very useful for identifying a power law (which it converts to a straight line) and I would like to suggest that showing an example of this (perhaps with a non-integer exponent) might be a worthwhile addition to your page - especially since the loglog transformation of an exponential (which you demonstrate) is again an exponential so the only effect in that case is an overall scaling.
cheers,
Alan
I love your blog entry on the use of a semilog graph to emphasize the area of greatest interest in a ranking situation and will be using it in my precalculus class a week or so from now.
I can also imagine that a loglog graph might be useful to get a view of high frequency small amplitude oscillations near the origin and/or big slow variations far away, but would be hard put to come up with such a topical example.
Of course a loglog graph is also very useful for identifying a power law (which it converts to a straight line) and I would like to suggest that showing an example of this (perhaps with a non-integer exponent) might be a worthwhile addition to your page - especially since the loglog transformation of an exponential (which you demonstrate) is again an exponential so the only effect in that case is an overall scaling.
cheers,
Alan
Thanks for the mail and I liked your suggestion which I implemented.
I added what you wrote as a comment on the blog. Hope it's okay with you.
Murray
X
Hi Alan
Thanks for the mail and I liked your suggestion which I implemented.
I added what you wrote as a comment on the blog. Hope it's okay with you.
Murray
I really enjoyed reading your thoughts on this, and I’m glad you brought it up because it touches on something a lot of people overlook when they talk about “just graphing the data.” The choice of scale can completely change what patterns jump out, and the semilog example on that page is one of the clearest demonstrations of that. I remember the first time I saw a ranking curve plotted on both linear and semilog axes — it suddenly made sense why so many systems (website traffic, city populations, even word frequencies) look chaotic on a normal plot but become remarkably well-behaved when you scale one axis.
Using it in a precalc class is a great idea. Students usually only see log scales in isolated textbook examples, but this type of visual explanation shows why the transformation is useful, not just how to compute it. It’s exactly the kind of thing that can give them an “aha” moment.
Regarding the log-log suggestion, I think you’re absolutely right that it deserves a place on that page. Log-log graphs have this almost magical ability to uncover structure in data that otherwise looks like noise. Anything that follows a power law suddenly becomes a straight line, which is incredibly satisfying to see. And showing a non-integer exponent — something like a decay of x^(-1.37) or a growth curve like x^(2.4) — would highlight the practical benefit. Students often think exponents have to be “nice,” so seeing fractional ones behave so cleanly on a log-log plot can broaden their intuition.
As for your idea of using a log-log graph to highlight high-frequency, small-amplitude oscillations near the origin versus slow, large-scale changes farther out — that’s a really intriguing angle. It’s true that finding a topical, real-world example might take some digging. Off the top of my head, the closest categories I can think of are:
• Certain types of fractal or self-similar signals, which naturally display very different behaviors at small vs. large scales.
• Some physical phenomena where amplitude shrinks but frequency increases as a parameter approaches zero — things like damping curves or specific types of resonance.
• Noise spectra (like 1/f noise) that can look messy normally but line up more neatly on log-log axes.
But even if it’s hard to find a “perfect” real-world example, I agree that demonstrating how log-log scales reveal or suppress different aspects of a function would be a useful addition.
And your point about the exponential transformation is spot-on: when you switch both axes to log, exponentials don’t get straightened out the way power laws do — they stay exponential, just rescaled. That contrast alone would make a nice educational comparison on the page, since it shows how different types of growth behave under different lens choices. brainrot clicker
X
I really enjoyed reading your thoughts on this, and I’m glad you brought it up because it touches on something a lot of people overlook when they talk about “just graphing the data.” The choice of scale can completely change what patterns jump out, and the semilog example on that page is one of the clearest demonstrations of that. I remember the first time I saw a ranking curve plotted on both linear and semilog axes — it suddenly made sense why so many systems (website traffic, city populations, even word frequencies) look chaotic on a normal plot but become remarkably well-behaved when you scale one axis.
Using it in a precalc class is a great idea. Students usually only see log scales in isolated textbook examples, but this type of visual explanation shows why the transformation is useful, not just how to compute it. It’s exactly the kind of thing that can give them an “aha” moment.
Regarding the log-log suggestion, I think you’re absolutely right that it deserves a place on that page. Log-log graphs have this almost magical ability to uncover structure in data that otherwise looks like noise. Anything that follows a power law suddenly becomes a straight line, which is incredibly satisfying to see. And showing a non-integer exponent — something like a decay of x^(-1.37) or a growth curve like x^(2.4) — would highlight the practical benefit. Students often think exponents have to be “nice,” so seeing fractional ones behave so cleanly on a log-log plot can broaden their intuition.
As for your idea of using a log-log graph to highlight high-frequency, small-amplitude oscillations near the origin versus slow, large-scale changes farther out — that’s a really intriguing angle. It’s true that finding a topical, real-world example might take some digging. Off the top of my head, the closest categories I can think of are:
• Certain types of fractal or self-similar signals, which naturally display very different behaviors at small vs. large scales.
• Some physical phenomena where amplitude shrinks but frequency increases as a parameter approaches zero — things like damping curves or specific types of resonance.
• Noise spectra (like 1/f noise) that can look messy normally but line up more neatly on log-log axes.
But even if it’s hard to find a “perfect” real-world example, I agree that demonstrating how log-log scales reveal or suppress different aspects of a function would be a useful addition.
And your point about the exponential transformation is spot-on: when you switch both axes to log, exponentials don’t get straightened out the way power laws do — they stay exponential, just rescaled. That contrast alone would make a nice educational comparison on the page, since it shows how different types of growth behave under different lens choices.
<a href="https://brainrot-clicker.io/">brainrot clicker</a>
Alan, your insights into the potential applications of loglog graphs are intriguing. The idea of visualizing high frequency small amplitude oscillations near the origin and big slow variations far away is fascinating. I agree that showcasing an example of a power law with a non-integer exponent on a loglog graph could be a valuable addition. It would provide a clear illustration of how loglog graphs can transform exponential functions into straight lines, especially emphasizing the scaling effect. Your upcoming precalculus class sounds engaging with the use of semilog graphs to highlight Snow Rider 3D areas of interest. Best of luck with your lesson!
X
Alan, your insights into the potential applications of loglog graphs are intriguing. The idea of visualizing high frequency small amplitude oscillations near the origin and big slow variations far away is fascinating. I agree that showcasing an example of a power law with a non-integer exponent on a loglog graph could be a valuable addition. It would provide a clear illustration of how loglog graphs can transform exponential functions into straight lines, especially emphasizing the scaling effect. Your upcoming precalculus class sounds engaging with the use of semilog graphs to highlight <a href="https://snowrider3d.com">Snow Rider 3D</a> areas of interest. Best of luck with your lesson!
Alan, your insights into the potential applications of loglog graphs are intriguing. The idea of visualizing high frequency small amplitude oscillations near the origin and big slow variations far away is fascinating. I agree that showcasing an example of a power law with a non-integer exponent on a loglog graph could be a valuable addition. It would provide a clear illustration of how loglog graphs can transform exponential functions into straight lines, especially emphasizing the scaling effect. Your upcoming precalculus class sounds engaging with the use of semilog graphs to highlight Snow Rider 3D areas of interest. Best of luck with your lesson!
X
Alan, your insights into the potential applications of loglog graphs are intriguing. The idea of visualizing high frequency small amplitude oscillations near the origin and big slow variations far away is fascinating. I agree that showcasing an example of a power law with a non-integer exponent on a loglog graph could be a valuable addition. It would provide a clear illustration of how loglog graphs can transform exponential functions into straight lines, especially emphasizing the scaling effect. Your upcoming precalculus class sounds engaging with the use of semilog graphs to highlight <a href="https://snowrider3d.com">Snow Rider 3D</a> areas of interest. Best of luck with your lesson!
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