Why the natural logarithm are not as confusing as you think.

Why the natural logarithm are not as confusing as you think.

This is my last post for the year 2020, by tomorrow Jan 1st, 2021, we are going to be marking a new decade, a decade of hope, unity, peace and progress.

Today i will be discussing on a very important and rather "confusing" area of mathematics especially to students studying the foundations of mathematics which is the natural logarithm function. I remember studying it in high school and how i so much disliked seeing it in exercises because of its confusing nature.

So i will be writing on "Why the natural logarithm are not as confusing as you think".

The natural logarithm of a number is defined as its logarithm to the base of the mathematical constant $e$ where $e$ is an irrational and a transcendental number approximately equal to $2.718281828459$. 

The natural logarithm of a number say $x$ can be written in the following ways:

1) $\ln{x}$ or $\log_e{x}$. 

2) $\log_e{(x)}$ with addition of parentheses for clarity. 

Now, the ralationship between the $\ln$ and $\log$ functions is another confusing aspect, in high school, you were probably told they were the same and can be used interchangeably, NO it is not TRUE. 

A logarithm is a form of function used to solve the following type of problems:

$a^x=b$, this is simply saying, what power do i need to raise $a$ to in order to obtain $b$.

This can be denoted in logarithm as $\log_{a}b=x$. 

That value $a$ is what we call our base, and it can vary depending on what problem you're trying to solve. But when you have a base $10$, then it's conventional to just drop the base from the notation, since the base is defined as $10$ therefore, if we are to write $\log_{10}5$ then we just write it as $\log{5}$. 

But when the base becomes a Euler number $e$ then it becomes a $\ln$ function and if you must write it in terms of logarithm then it must be $\log_e$ for example $\ln(3)=\log_{e}3$. 

Sometimes the natural logarithm of $x$ written as $\ln{x}$ is written as $\log(x)$, we must understand that, in this case, it was not written interchangeably, it was only written knowing that $e$ is implicit. 

Below is a graph of the natural logarithm function. The function slowly grows to positive infinity as $x$ increases, and slowly goes to negative infinity as $x$ approaches $0$ ("slowly" as compared to any power law of $x$); the $y$-axis is an asymptote.

In a much simpler language, we can define the natural logarithm of a number say $x$ as the power to which $e$ would have to be raised to equal $x$ e.g if we compute $\ln{2}$ it equals $0.69314718...$ and if we compute $e^{0.69314718055995}=2.00000000000001$ and the natural logarithm of $e$ itself, $\ln{e}$ is $1$, because $e^1=e$, while the natural logarithm of $1$ is $0$, since $e^0=1$. 

The natural logarithm is also defined for any positive real number $a$ as the area under the curve $y=\frac{1}{x}$ from $1$ to $a$ with the area being negative when $0

The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:

$e^{\ln{x}}=x$ for $x>0$ and 

$\ln{e^x}=x$ for $x>0$.

SOME PROPERTIES OF THE NATURAL LOGARITHM FUNCTIONS. 

Below are some properties of the natural logarithm functions that will help you in evaluating problems associsted to natural logarithms. 

1) Multiplicative property: $\ln(xy)=\ln{x}+\ln{y}$ $\forall{x,y>0}$

2) $\ln{1}=0$

3) $\ln{e}=1$

4) $\ln{x^y}=y\ln{x}$ for $x>0$

5) $\ln{x}<\ln{y}$ for $0

6) $\lim_{x\rightarrow{0}}\frac{\ln(1+x)}{x} =1$

7) $\lim_{a\rightarrow{0}}\frac{\ln(x^a-1)}{a}=\ln{x}$ for $x>0$

8) $\frac{x-1}{x}\leq{\ln{x}}\leq{x-1}$ for $x>0$

9) $\ln(1+x^a)\leq{ax}$ for $x\geq{0}$ and $a\geq{1}$.

10) Derivative of natural logarithm: $\frac{d}{dx}\ln{x}=\frac{1}{x}$ 

11) $\int\frac{1}{x}dx=\ln|x|+C$

12) Integral of natural logarithm: $\int{\ln{x}}dx=x\ln{x}-x+C$

13) Series properties: $\ln(1+x)=\sum^{\infty}_{k=1}\frac{(-1)^{k-1}}{k}x^k=x-\frac{x^2}{2}$ for $|x|\leq{1}$ and $x\neq{-1}$

14) Natural logarithm of $10$: $\ln(a.10^n)=\ln{a}+n\ln{10}$