Question

How do you simplify $\ln \left( 1-{{e}^{-x}} \right)$

Answer 1

Hint: Now to simplify the given expression we will first write ${{e}^{-x}}=\dfrac{1}{{{e}^{x}}}$ . Now we will take LCM in the function and try to simplify the term $1-\dfrac{1}{{{e}^{x}}}$ . Now after simplifying the obtained expression we will use the division rule of logarithm which says $\log \left( \dfrac{p}{q} \right)=\log p-\log q$ . Now again we will use the exponent rule $\ln {{a}^{n}}=n\ln a$ . Now we know that $\ln e=1$ hence we will substitute the value of $\ln e$ and hence simplify the expression.

Complete step by step solution:
Now we are given an expression in ln.
Let us first consider the given expression $\ln \left( 1-{{e}^{-x}} \right)$
Now we can rewrite the expression as $\ln \left( 1-\dfrac{1}{{{e}^{x}}} \right)$
Now simplifying the expression we get, $\ln \left( \dfrac{{{e}^{x}}-1}{{{e}^{x}}} \right)$
Now we know that according to division rule of logarithm $\ln \left( \dfrac{a}{b} \right)=\ln a-\ln b$
Hence using this we get,
$\Rightarrow \ln \left( {{e}^{x}}-1 \right)-\ln \left( {{e}^{x}} \right)$
Now we know that according to exponent law of logarithm ${{\log }_{a}}{{x}^{n}}-n{{\log }_{a}}x$ , hence using this we get,
$\Rightarrow \ln \left( {{e}^{x}}-1 \right)-x\ln \left( e \right)$
Now we know that ln is nothing but a logarithm with base e and for any positive number a not equal to 1 we have ${{\log }_{a}}a=1$ . Hence we get
$\Rightarrow \ln \left( {{e}^{x}}-1 \right)-x\ln \left( {{e}^{x}} \right)=\ln \left( {{e}^{x}}-1 \right)-x\left( 1 \right)$

Hence the given expression can be written as $\ln \left( {{e}^{x}}-1 \right)-x$

Note: Now note that we have subtraction inside the function of logarithm and hence we have no rule for this. Do not confuse with the division rule $\ln \left( \dfrac{a}{b} \right)=\ln a-\ln b$ . Now note that ln is called the natural logarithm and is nothing but a logarithm with base e where e is 2.7182…
Now for logarithm remember that the value inside the logarithm must be always positive and the base must be a positive number which is not equal to 1.