Question

How do you solve $\ln x = 3$?

Answer 1

Hint: The function $y = {e^x}$and $y = \ln x$are inverse functions. To solve this problem, we need to know the logarithmic and radical properties which are as follows,
${e^{\ln x}} = x$ and also, $\ln {e^x} = x$.

Complete step by step solution:
According to the given information, we have to solve $\ln x = 3$.
We know that the natural log ($\ln $) and the exponential function (${e^x}$) are inverses of each other.
We need to be clear about the fact that if something is done on one side of the equation, the same thing has to happen on the opposite side of the equation too.
In this case, if we raise $e$ to the $\ln x$, we are just left with x on the left side as we know ${e^x}$ and $\ln $ undo each other,
\[{{e}^{{{\ln x}}}} = 3\]
Now, we have to do the same thing on the right side and raise e to the third power like this, we get
$x = {e^3}$
When you do the calculation of the above expression, you obtain an approximate value of $20.09$.

Thus, $x = 20.09$

Note:
We know that the natural log ($\ln $) and the exponential function (${e^x}$) are inverses of each other, which means that if we raise the exponential function by $\ln $ of $x$, we would be able to find x. But remember, if something is done on one side of the equation, the same thing has to happen on the opposite side of the equation too.