<p>Generally <code>ln</code> is used to represent the <a href="https://en.wikipedia.org/wiki/Natural_logarithm" rel="nofollow noreferrer">natural logarithm</a>.  I know of no notation system that uses it to represent differentiation.  </p>

<p>Natural logarithm:  </p>

<p><span class="math-container">$$\frac{\ln{y} - \ln{y_0}}{t}$$</span></p>

<p><a href="https://en.wikipedia.org/wiki/Differential_calculus" rel="nofollow noreferrer">Differentiation</a> would look more like </p>

<p><span class="math-container">$$\frac{dy}{dt}$$</span></p>

<p>or </p>

<p><span class="math-container">$$f'(t)$$</span></p>

<p>It is common to take <a href="https://en.wikipedia.org/wiki/Logarithm" rel="nofollow noreferrer">logarithms</a> of growth functions because they involve exponents and logarithms can simplify equations with exponents.  In particular, </p>

<p><span class="math-container">$$ \ln{(1+g)^t} = t\ln{(1+g)}$$</span></p>

<p>is a basic property of logarithms.  </p>

<p>A Maclaurin expansion is an alternative way to write the equation.  See <a href="https://en.wikipedia.org/wiki/Taylor_series" rel="nofollow noreferrer">Taylor series</a> and <a href="http://www.mathsmutt.co.uk/files/macser.htm" rel="nofollow noreferrer">Maclaurin's series</a> for more information.  A Maclaurin expansion does in fact take advantage of differentiation.  </p>

<p>This logarithm is referred to as natural because its base arises from normal mathematical calculations rather than being arbitrary, much like <span class="math-container">$\pi$</span>.  </p>

CC BY-SA 4.0
2018-09-23T02:08:25.050Z
Is there any intuition of using (lny-lny0)/t?
user19506
CC BY-SA 4.0
2018-09-23T01:53:41.163Z
2018-09-23T01:53:41.163Z
Brythan
<p>I  cannot understand what (lny-lny0)/t means. Of course we use this to show growth rate. But in the mathematical sense, I cannot know what it is. Is it diferrentiaion?</p>

CC BY-SA 4.0
2018-09-22T15:37:25.737Z
Hi and welcome tk Economics Stack Exchange. Can you please give us some more context. Where did you find the expression and what was it used for there?
BB King
CC BY-SA 4.0
2018-09-22T15:39:01.063Z
to find out growth rate, first, y=y0(1+g)^t and lny=lny0+tln(1+g) and (lny-lny0)/t = ln(1+g) using maclaurin expansion ln(1+g) = g so (lny-lny0)/t = g
user19506
CC BY-SA 4.0
2018-09-22T15:40:57.730Z
Growth rate of what? What is t, what is g, what is y? How did the author you saw this from explain it? Please add the info in your comment and all other info to the main text of the question.
BB King
CC BY-SA 4.0
2018-09-22T15:44:04.913Z
Oh i edit my explanation.
user19506
CC BY-SA 4.0
2018-09-22T17:31:09.320Z
It would help if you edited the question using the math markup that is available on this site, and to maybe write out some of these formulas you have in the comments up there.
Kitsune Cavalry
<economic-growth>
I can't find (lny-lny0)/t meaning
user19506
# I can't find (lny-lny0)/t meaning

(Asked by: username_0 on 2018-09-22)

I cannot understand what (lny-lny0)/t means. Of course we use this to show growth rate. But in the mathematical sense, I cannot know what it is. Is it diferrentiaion?

**Tags:** economic-growth

> **Comments:**
> **username_1**: Hi and welcome tk Economics Stack Exchange. Can you please give us some more context. Where did you find the expression and what was it used for there?  
> **username_0**: to find out growth rate, first, y=y0(1+g)^t and lny=lny0+tln(1+g) and (lny-lny0)/t = ln(1+g) using maclaurin expansion ln(1+g) = g so (lny-lny0)/t = g  
> **username_1**: Growth rate of what? What is t, what is g, what is y? How did the author you saw this from explain it? Please add the info in your comment and all other info to the main text of the question.  
> **username_0**: Oh i edit my explanation.  
> **username_2**: It would help if you edited the question using the math markup that is available on this site, and to maybe write out some of these formulas you have in the comments up there.  

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## 1 answers

### Answer by username_3 on 2018-09-23. Score: 1 | Accepted answer

Generally `ln` is used to represent the [natural logarithm](https://en.wikipedia.org/wiki/Natural_logarithm). I know of no notation system that uses it to represent differentiation.

Natural logarithm:

$$\frac{\ln{y} - \ln{y\_0}}{t}$$

[Differentiation](https://en.wikipedia.org/wiki/Differential_calculus) would look more like

$$\frac{dy}{dt}$$

or

$$f'(t)$$

It is common to take [logarithms](https://en.wikipedia.org/wiki/Logarithm) of growth functions because they involve exponents and logarithms can simplify equations with exponents. In particular,

$$ \ln{(1+g)^t} = t\ln{(1+g)}$$

is a basic property of logarithms.

A Maclaurin expansion is an alternative way to write the equation. See [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) and [Maclaurin's series](http://www.mathsmutt.co.uk/files/macser.htm) for more information. A Maclaurin expansion does in fact take advantage of differentiation.

This logarithm is referred to as natural because its base arises from normal mathematical calculations rather than being arbitrary, much like $\pi$.

> **Comments:**
> **username_0**: Is there any intuition of using (lny-lny0)/t?  

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