# Math Trouble in Physics Land: Logarithms

• Chrono G. Xay
In summary, the conversation discusses a personal music project and the difficulty of pulling 'z' out of a logarithm in a quadratic model. The physics aspect of the project is also mentioned, and there is a discussion about solving for 'z' using different equations and methods. The conversation also touches on strain, initial segments and deformations. The speaker expresses a desire to minimize the use of subscripts in their equations for visual clarity.
Chrono G. Xay
Working on a personal music project, I would like to pull 'z' out of the logarithm below if I can help it, but am having trouble:

It's a portion taken from this:

Which evolved from...

However, this is only the mathematical modeling of the object. The actual *physics* aspect is much more simple, and has already been worked out from a linear model I developed before it. The above quadratic model is more realistic.

Anyway, my hope is that if I can pull 'z' out of the logarithms I can pull it all the way out of the second integral (theta). If not- I'll just have to see what I can do. (I don't have my spiral with me to check.)

What do you mean by "pull z out of the logarithm"?
You don't have an equation to solve for z.

But even if you have, it is quite likely that there is no way to solve it algebraically.

Factor it out.

You cannot factor the z from that expression - it is impossible to remove it from the logarithmic expressions.

Ok. I expected that was the case, but not having that much 'collegiate experience' in math I wanted to ask, anyway. Thank you.

You can't remove ##z## completely from the logarithm expression, but you can solve for it:
##S = \ln\Big(\sqrt{\frac{z^2+(2L_n)^2}{2L_n}}+2\sqrt{L_n}\Big)-\ln\Big(\sqrt{\frac{z^2+(2L_n)^2}{L_n}}-2\sqrt{L_n}\Big)\\
=2\ln(\sqrt{z^2+(2L_n)^2}+2L_n)-2\ln(z)=2\ln\Big(\sqrt{1+(\frac{2L_n}{z})^2}+\frac{2L_n}{z}\Big)=2\operatorname{arsinh}\Big(\frac{2L_n}{z}\Big)\\
\Rightarrow z=\frac{2L_n}{\sinh(S)}.##​

nasu
suremarc said:
##\Rightarrow z=\frac{2L_n}{\sinh(S)}.##
My bad--that should be ##z=\frac{2L_n}{\sinh(\frac{S}{2})}.##

That solution does nothing for the original question.

The reason I was hoping I'd be able to factor 'z' to where it occurred only once in the equation and was multiplied by everything else (held inside a likely large parentheses) was so I could use it as the spring displacement in the equation for spring force, where, obviously, everything else was the spring constant (of a circular membrane) in a mass-spring-damper system. Then again, how does one go about mathematically describing such a system when the mass is, itself, the spring?

You have some very strange things in your posts. In your fifth image you have
$$\frac{\Delta L}{L}= \frac{L- L}{L}= \frac{L}{L}- 1= \frac{L_1- L_2}{L}- 1$$
None of that makes any sense at all!

ΔL / L_0 = ( L_f - L_0 ) / L0 = ( L_f / L_0 ) - ( L_0 / L_0 ) = ( L_f / L_0 ) - 1

What's strange about that? It's the equation for strain, ε.

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Okay- I think I see the problem here. I expressed 'L_0' as a lowercase 'l' for the sake of minimizing the use of subscripts, and as such I made sure to make the curve at the bottom of the 'l' pronounced.

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As for l_1 and l_2, I was adding two distinct initial chord segments (not to be confused with chord lengths, as c=2*sqrt(r^2-d^2) can be expanded to equal the sum of two line segments, sqrt(r^2-d^2)+sqrt(r^2-d^2) ). L_1 and L_2 are the initial segments, being deformed and, thus, stretched to a final length.

HallsofIvy said:
You have some very strange things in your posts. In your fifth image you have
$$\frac{\Delta L}{L}= \frac{L- L}{L}= \frac{L}{L}- 1= \frac{L_1- L_2}{L}- 1$$
None of that makes any sense at all!
##L_n, l_n## and##\ ln## present an inscrutable mix in modified-cursive handwritten font!

I see it as ##\frac{\Delta L}{l} = \frac{L - l}l##

I feel confused by your reply... I never used "l_n" or "L_n"... I used l_1 and l_2, and L_1 and L_2... natural log was always written as 'ln(x)'...

L ( AKA "L_f" ) = L_1 + L_2
l ( AKA "L_0" ) = l_1 + l_2

Looking back still farther, I think I see what you were referring to (which I interpreted to be "z * sqrt( x / l_n )" ). Yeah, I saw how that could be perceived as ambiguous. I still wanted to try to minimize subscripts to make it more visually intuitive, especially considering the type of system being described.

## 1. What are logarithms and how are they used in physics?

Logarithms are mathematical functions used to solve exponential equations. In physics, they are commonly used to express large or small quantities in a more manageable form. They also help to transform complex mathematical equations into simpler forms, making calculations easier.

## 2. Why do logarithms cause trouble in physics?

Logarithms can cause trouble in physics because they involve complex functions and calculations, which can lead to errors if not approached carefully. If used incorrectly, they can cause significant discrepancies in results and make it difficult to interpret data accurately.

## 3. What are some common mistakes people make when dealing with logarithms in physics?

One common mistake is forgetting to convert units when using logarithms, which can lead to incorrect results. Another mistake is using the wrong base for the logarithm, as different bases can give different results. It's also important to properly apply the rules of logarithms, as even small errors can have a significant impact on the final answer.

## 4. How can I improve my understanding and application of logarithms in physics?

Practicing and reviewing logarithm problems regularly can help improve your understanding and application. It's also helpful to understand the properties and rules of logarithms, as well as their applications in physics. Seeking help from a tutor or teacher can also provide valuable guidance and clarification.

## 5. Are there any alternative methods to using logarithms in physics?

Yes, there are alternative methods to using logarithms in physics, such as using scientific notation or performing calculations with decimal powers. However, logarithms are still widely used and can often provide more efficient and accurate results in certain situations.

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