<p>First of all you must have some faulty calculator or you are using wrong base. Natural log of 2 is about 0.7 not 0.3.</p>
<p>Small here means close to zero. If Age is 0 then <span class="math-container">$\ln(1+0)=0$</span>. If Age is 0.5 (6 month old baby) the <span class="math-container">$\ln (1+0.5)= 0.41$</span></p>
<p>This is because for values of <span class="math-container">$x$</span> close to zero <span class="math-container">$\ln (1+x) \approx x$</span>. But this will give you fairly good approximation only when <span class="math-container">$x&lt;1$</span>. For <span class="math-container">$x&gt;1$</span> the error between true value and approximation quickly increases. For example <span class="math-container">$\ln (1+1) =0.7$</span>, <span class="math-container">$\ln (1+2) =1.1$</span> at this point the error is already quite large.</p>

CC BY-SA 4.0
2021-10-27T22:48:25.840Z
Ah I see, thanks @1muflon1, very easy to follow. First of all, normally, I studies that log mean log10, not loge (ln), therefore I quite misunderstand. Secondly, it is now clear to me that "small" means "lower than 1" in this case.
Phil Nguyen
CC BY-SA 4.0
2021-10-27T23:55:31.083Z
typically in econometrics and math nowadays people use log meaning ln even though you are right by old convention $\log=\log_{10}$
1muflon1
CC BY-SA 4.0
2021-10-27T22:17:52.243Z
2021-10-30T07:13:50.367Z
2021-10-30T07:13:50.367Z
1muflon1
<p>I see a conversation <a href="https://quant.stackexchange.com/questions/68534/why-we-should-use-log1age-rather-than-logage?noredirect=1#comment98504_68534">here</a> that @dm63 mention that</p>
<blockquote>
<p>when Age is small, log(1+Age)=Age</p>
</blockquote>
<p>I am wondering how can it happen. For example, when saying Age is small, I chose Age =1 or 2</p>
<p><strong>Age =1 =&gt; log(1+1)=log(2)=0.3 # 1</strong></p>
<p><strong>Age =2 =&gt; log(1+2)=log(3)=0.477 # 2</strong></p>
<p>I may fall into a fallacy, could you please help me to sort it out?
Much appreciated.</p>

CC BY-SA 4.0
2021-10-27T22:40:36.493Z
the reason why $\ln(1+x) \approx x $ when $x$ is close to $0$ is that $x$ is the the taylor polynomial of order $1$ of $\ln(1+x)$
Tortar
<econometrics>
Why when a variable $x$ is small, $\log (1+ x) = x$?
Phil Nguyen
# Why when a variable $x$ is small, $\log (1+ x) = x$?

(Asked by: username_0 on 2021-10-27)

I see a conversation [here](https://quant.stackexchange.com/questions/68534/why-we-should-use-log1age-rather-than-logage?noredirect=1#comment98504_68534) that @dm63 mention that

> when Age is small, log(1+Age)=Age

I am wondering how can it happen. For example, when saying Age is small, I chose Age =1 or 2

**Age =1 => log(1+1)=log(2)=0.3 # 1**

**Age =2 => log(1+2)=log(3)=0.477 # 2**

I may fall into a fallacy, could you please help me to sort it out?
Much appreciated.

**Tags:** econometrics

> **Comments:**
> **username_1**: the reason why $\ln(1+x) \approx x $ when $x$ is close to $0$ is that $x$ is the the taylor polynomial of order $1$ of $\ln(1+x)$  

---

## 1 answers

### Answer by username_2 on 2021-10-27. Score: 2 | Accepted answer

First of all you must have some faulty calculator or you are using wrong base. Natural log of 2 is about 0.7 not 0.3.

Small here means close to zero. If Age is 0 then $\ln(1+0)=0$. If Age is 0.5 (6 month old baby) the $\ln (1+0.5)= 0.41$

This is because for values of $x$ close to zero $\ln (1+x) \approx x$. But this will give you fairly good approximation only when $x<1$. For $x>1$ the error between true value and approximation quickly increases. For example $\ln (1+1) =0.7$, $\ln (1+2) =1.1$ at this point the error is already quite large.

> **Comments:**
> **username_0**: Ah I see, thanks @username_2, very easy to follow. First of all, normally, I studies that log mean log10, not loge (ln), therefore I quite misunderstand. Secondly, it is now clear to me that "small" means "lower than 1" in this case.  
> **username_2**: typically in econometrics and math nowadays people use log meaning ln even though you are right by old convention $\log=\log_{10}$  

---

