Question

How do you solve $\ln x = - 7?$

Answer 1

Let us understand in a general way, the function of a logarithm, Consider a logarithm of $a$ with base $b$ equal to $c$ , it means that when $c$ is raised to the power of $b$ then it will equal to $a$ , let’s understand this in mathematical way or in equations $\Rightarrow {\log _b}a = c$ It is the mathematical form of the first part of the consideration. Now coming to the second part $\Rightarrow {b^c} = a$ It is the mathematical form of the second part. Hope that you got to understand the logarithmic function. $\ln$ is a special logarithmic function which has a fixed base equals $e$ Try to write the problem in equation form and then solve it further.

Complete step by step solution:
We have to find the value of $x$ using some properties logarithm and
exponential, Now let’s come to the question, we have $\ln x = - 7$ , we can also write this in $\log$ form as
${\log _e}x = - 7$ _____(I)

Logarithmic function has a property that if it is raised on the power of its base then only argument of the logarithm left, let us understand this with an example ${\log _a}b$ , if we raise ${\log _a}b$ in the power of $a$ then only $b$ will left
$\Rightarrow {a^{{{\log }_a}b}} = b$

Now we will raise both sides of equation (I) on the power of $e$ , we will get
$\Rightarrow {e^{{{\log }_e}x}} = {e^{ - 7}} \\\ \Rightarrow x = {e^{ - 7}} \\\$

We got the required solution for $\ln x = - 7$ which is $x = {e^{ - 7}}$
If you want numerical value then calculate this on a calculator you will get a value close to $x = 9.12 \times {10^{ - 4}}$

Note: The functions $y = {e^x}\;\& \;y = \ln x$ are inverse functions of each other. Base of a logarithm function should be greater than zero and can’t be equal to one.
Argument of the logarithm function is always positive.