Question

How do you solve \[2\;\ln (x) = 1\]?

Answer 1

Hint:The logarithm is the inverse to exponentiation. meaning the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to supply that number x. Within the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication.

Complete step by step answer:
Using BODMAS rule, we see that there are no brackets, so we have division that we can undo to isolate the \[\ln (x)\],
Dividing both sides of the equation by 2,
\[2\;\ln (x) = 1\]
$ \Rightarrow \dfrac{{2\;\ln (x)}}{2} = \dfrac{1}{2}$
$ \Rightarrow \ln (x) = \dfrac{1}{2}$
Now ln(x) has been isolated, we exponentiate the equation.
\[ \Rightarrow {e^{lnx}} = {e^{\dfrac{1}{2}}}\]
Exponential and base-e log are inverse functions.
\[ \therefore x = {e^{\dfrac{1}{2}}}\]

Note: Logarithms are vastly used to form logarithmic models to analyse data.Logarithmic models are used to measure the magnitude of the thing we need to measure. It can also be seen as the inverse of an exponential model. For example, exponential growth is very common in nature for things like radioactivity, bacterial growth, etc., being written as
\[N(t) = {N_0}{e^{kt}}\;or\;N(t) = {N_0}{a^t}\]