Question

Is $\left({ln\ x} \right)^{2}$ equivalent to $ln^{2}\left( x \right)$ ?

Answer 1

In this question, we need to find whether $\left({ln\ x} \right)^{2}$ is equivalent to $ln^{2}\left( x \right)$ . ln is nothing but it is a natural log that is log with base $e$. $log{_e}= \ ln$ (natural log). Here $e$ is the exponential function. Mathematically, ln is known as natural logarithm and it is also known as logarithm of base $e$ . Mathematically it is represented as ${ln\ x}$ or $\log{_e} x$ . Logarithm is nothing but a power to which numbers must be raised to get some other values. Here we need to find whether $\left({ln\ x} \right)^{2}$ is equivalent to $ln^{2}\left( x \right)$ .
Logarithmic property used :
${log\ }\left( m^{n} \right) = \ n\ log\ m$

Complete step-by-step solution:
We need to find whether $\left( {ln\ x} \right)^{2}$ is equivalent to $ln^{2}\left( x \right)$
In order to find whether $\left( {ln\ x} \right)^{2}$ and $ln^{2}\left( x \right)$ are equivalent, first we can assume that $ln^{2}\left( x \right) = \left( {ln\ x} \right)^{2}$
Let us assume that
$ln^{2}\left( x \right) = \left( {ln\ x} \right)^{2}$
From the property of logarithm,
${log\ }\left( m^{n} \right) = \ n\ log\ m$
We get,
$\Rightarrow (\ln{\ x)}^{2} = 2\ln x$
$(\ln{\ x)}^{2}$ is $ln\ x \times ln\ x$ Since $ln\ x\ \neq 0$
$\Rightarrow ln\ x \times ln\ x = 2\ln x$
On dividing both sides by $ln\ x$ ,
We get,
${ln\ x} = 2$
On taking exponential on both sides we get,
$x = e^{2}$
Hence we can conclude that $ln^{2}\left( x \right) = \left( {ln\ x} \right)^{2}$ is true for $x = e^{2}$
Final answer :
$\left({ln\ x} \right)^{2}$ is equivalent to $ln^{2}\left( x \right)$

Note: We need to know that the logarithmic function to the base $e$ is known as the natural logarithmic function and it is denoted by $\log{_e}$. We should not be confused ${ln\ }x^{2}$ with $2ln\ x$ where both are the same. The inverse of logarithm is known as exponential. Exponential function is nothing but it is a mathematical function which is in the form of $f\ (x)\ = \ a^{x}$, where $x$ is a variable and $a$ is a constant . The most commonly used exponential base is $e$ which is approximately equal to $2.71828$ .
Few properties of logarithm are
1.$log\ mn\ = \ log\ m\ + \ log\ n$
2.${log\ }\left( \dfrac{m}{n} \right) = \ log\ m\ \ log\ n$
3.${log\ }\left( m^{n} \right) = \ n\ log\ m$