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 $10$: $\ln(a.10^n)=\ln{a}+n\ln{10}$
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