Question

How do we solve ${e^{ - 3n}} = 83$ ?

Answer 1

Hint:To solve this question, first we should think about removing the term $e$ from the given equation. To remove the term $e$ , take the natural log of both sides. And, in this way we will get the value of $n$ .

Complete step by step solution:
The given equation is as:
As we know, $\ln (x)$ is the natural logarithm or ${\log _e}(x)$ :
Now, we can take the natural log of both sides to get $e$ out of the equation:
$\ln ({e^{ - 3n}}) = \ln (83)$
Since the $\ln $ and the cancel out and we get:-
$ \Rightarrow - 3n = \ln (83)$
$\therefore n = - \dfrac{{\ln (83)}}{3} \approx - 1.473$
Hence, in the given equation- the value of is equals to $ - 1.473$ .

Note:- The exponent of the base of $\ln $ which gives us the integrand, ${e^x}$ ;
So, the base of $\ln $ is $e$ ; The number we need to be the exponent of this base to get is.....exactly $x$ !
So: \[ln({e^x}) = lo{g_e}({e^x}) = x\] .