Question

How do I find a natural log of a fraction $?$

Answer 1

Hint: The natural log is defined as $\ln x$ is the logarithm having a base $e$, where
$e=2.718281828.......$
The function also defines as \[\ln x=\int\limits_{1}^{x}{\dfrac{dt}{t}}\] and $\ln x={{\log }_{e}}x$.
Rules and properties of natural logarithm are
Product rule of logarithm $\ln (x.y)=\ln x+\ln y$
Quotient rule of logarithm $\ln \left( \dfrac{x}{y} \right)=\ln x-\ln y$
Power rule of logarithm $\ln ({{x}^{y}})=y.\ln x$

Complete step by step solution:
The natural logarithm of the fraction is defined as
\[\Rightarrow \]$\ln \left( \dfrac{x}{y} \right)=\ln x-\ln y$.
Now we will prove property using rules of natural logarithm for a fraction
We will take L.H.S of the above Quotient rule/property
$\Rightarrow \ln \left( \dfrac{x}{y} \right)=\ln (x{{y}^{-1}})$
$\Rightarrow \ln (x.{{y}^{-1}})=\ln x+\ln ({{y}^{-1}})$ $[$By using the product rule of the natural log$]$
$\Rightarrow \ln x+(-1.\ln y)$ $[$By using the power rule of the natural logarithm$]$
$\Rightarrow \ln x-\ln y$
$\Rightarrow $R.H.S of $\ln \left( \dfrac{x}{y} \right)$
Hence, now we can find any natural log of fraction for example
$\ln \left( \dfrac{1}{125} \right)=\ln 1-\ln 125=0-\ln {{5}^{3}}=-3\ln 5=-3$.

Note: The logarithm of the fraction is $=$ the logarithm of the numerator – logarithm of the denominator.
The natural logarithm of $1$ is $0$:$\ln 1=0$.
Base switch rule of logarithm ${{\log }_{b}}(c)=\dfrac{1}{{{\log }_{c}}(b)}$.
Base change rule of logarithm ${{\log }_{b}}(x)=\dfrac{{{\log }_{c}}(x)}{{{\log }_{c}}(b)}$.
The natural logarithm $0$ is undefined because the natural logarithm is always defined only for $x>0$.
The natural logarithm of the multiplication of $'x'$ and $'y'$ is equal to the sum of the natural log of $'x'$ and the natural log of $'y'$. It’s known as the product rule of the logarithm.