Question

How do you solve $\ln x = 4$?

Answer 1

As logarithm and exponentials are inverse functions of each other, we will take exponential on both sides of the given equation so as to cancel out the logarithm term and solve the question.

Formula used:
${\log _b}a = c$
$\Rightarrow a = {b^c}$
Also,
${e^{{{\log }_e}m}} = m$
Here, $a,b,c,m$ can be variables or constants.

Complete step-by-step answer:
We have to know that,
$\ln x = {\log _e}x$ , where $e$ is the Euler’s number.
($e \approx 2.72$).
So, the question can be written as,
$\ln x = {\log _e}x = 4$
$\Rightarrow {\log _e}x = 4$
Now, let us take exponents on both sides of the above equation. That is,
${e^{{{\log }_e}x}} = {e^4}$
(We have taken to power $e$ as the base of the logarithm is $e$)
We know that, ${e^{{{\log }_e}x}} = x$
Therefore,
$x = {e^4}$, which is our final answer.

Additional information:
$\ln$ , which is a logarithm to base $e$ is called the ‘Natural logarithm’.
${\log _{10}}$, which is normally written without the base as $\log$ is called the ‘Common logarithm’.
Trying to solve logarithmic equations graphically is a tedious task and is difficult to be done manually, except for very small values of $x$.
The basic operation rules for logarithm can also come in handy for solving those equations having more than one logarithm term.

Note: In order to take the reverse of any logarithm, the exponential must be taken with base same as that of the logarithm.